I have the same hang up with Race For The Galaxy - I like the game itself, and I play the fanmade PC version quite a bit, but when I play it with friends it’s really an autistic experience. Yes, you can check what the others are doing, how fast they’re going, gamble on which phase they’re going to play, mess with them by hanging onto key cards they need etc…, and the extensions have increased the interaction with the ability to steal planets from other players, but for the most part you can just as successfully focus on your own game only 
For a game which has a really interactive process, but also a ‘parallel soltatire’ phase, try Princes of Florence. It has the usual optimize whatever you have gameplay, but also a rather cutthroat but controlled auction phase. It’s interactive and great strategy at the same time.
How has this thread gone so far with no mention of Chaos in the Old World? If you don’t object to the theme (you play one of 4 Chaos Powers…think demon gods…who compete to corrupt and ruin the world) you’ll find a wonderful blend of worker placement, asymmetric powers, and simple wargame mechanics. And there are two entirely different ways to win, which adds an extra level of strategy to it.
Every time I’ve played, I’ve enjoyed it more, even the few times that I’ve been steamrolled.
I’d recommend Mystery of the Abbey and Acquire.
Victory Games, a division of Avalon Hill, put out some games that were designed for solo play. I had lots of fun with Ambush! and its expansions; also had Battle Hymn but couldn’t find time to play much. Mosby’s Raiders is an interesting concept but I have yet to get one.
Did I misunderstand a requirement or something… Axis and Allies is pretty much a turn in your wargamer card if you never played member of such lists.
Well I have a hunch that the OP is asking for recent games; he/she might already know off Axis and Allies.
That and A&A is a (light) wargame, not an ‘award-winning European strategy game’, two very different genres.
Actually Axis and Allies is a great suggestion. I have friends with copies and I enjoyed it immensely, but there are several expansions I haven’t played yet (the pacific one comes to mind).
My friend and I actually have developed a special rule-set for Twilight Struggle that we put to the test this past weekend. For those familiar with the game, we made a rule that you HAVE to play the cards in the order you received them, and you can’t look ahead in the deck. This turned the game from being based around playing the right card at the right time, and instead places the focus upon being able to manage your downfall as gracefully as possible. Trust me, it’s fun once you’ve tired of the base game and want to add a bit more challenge to it.
Obviously, there’s crossover in the audience for different game genres. But A&A isn’t what you asked for. 
I played a very interesting game last night called Here I Stand. It’s set during the Protestant Reformation in Europe. There are six powers: English, French, Hapsburgs/Holy Roman Empire, Protestants, Ottomans and the Papacy.
It is an extremely complicated game – the rulebook has 41 pages – although some of the complications are mitigated by the fact that not all players have to deal with all of the facets of the game. For example, the Ottomans can launch pirate raids in the Mediterranean Sea, but only the powers boarding the Med have to deal with it. The English, the French and the Holy Roman Empire can launch explorations or conquests of the New World, but the other three powers don’t interact with it at all. The Protestant player is trying to convert as many places as possible to Protestantism while the Papacy is trying to convert the new Protestants back to Catholicism, but the other powers don’t really care(although the English player does get a bonus for English cities that convert to Protestantism). And of course there’s a military aspect to the game as well.
I found that the game was reasonably well balanced despite the fact that every power can work towards their win condition in different way (some common with other powers, some unique to the player). Another thing that I liked is that the game really does a good job of preventing the problem of huge stacks just rolling over everybody until it runs into another huge stack. There are a lot of penalties and limits on stacks.
I loved the game, but I do have these warnings:
a) This game involves a significant time commitment, especially the first couple of times you play. Last night 5 of the 6 players had never played before and it took us over 11 hours to get through 4 turns(fortunately, somebody managed to win at that point). That’s very long for a game, but supposedly once you get good at the game you’re looking at 45 minutes per turn and a maximum of 9 turns, which is 6 hours and 45 minutes. Make sure that every player has at minimum read the sections of the rulebook that apply to the power that they will be playing ahead of time. This is not a game that you can really teach somebody from scratch at the table
b) The game really works much better with a full complement of 6 players. With fewer players some players will have to control 2 powers, and there can be a big advantage in handicapping one power to help the other power along towards an early victory. I really would suggest only playing with 3 or 6 players, and the 6 player game will be far better
c) I’ve only played it once, so I’m not sure how replayable the game is. One thing that will really help here is the fact that every power is so different from the others, which means that you could play the game six times, once with each power, and get a completely different game experience each time.
Actually, Axis & Allies offers enough strategic subtlety to compete with some of the well-designed strategy games like Puerto Rico and Settlers of Cataan…if you take it seriously despite the little plastic tanks and don’t just roll dice and complain about the results. The game is about marshaling resources and deciding when to commit to risk for rewards.
Be advised that people can be poor judges of dice probability. I’ve seen players make a bad attack in a bad place for a bad reason, roll perfectly average, and claim that the dice are against them. I’ve also seen things like someone complain that he rolled four sixes (resulting in four misses) and hold that up as an example of incredible bad luck – when he needed ones to hit, and four twos or above would also have missed. “Dude, it’s immaterial that they’re sixes…twos would have missed also,” doesn’t win an argument with someone moaning over how unlikely four consecutive sixes are.
Although I traditionally have disliked bidding games, I found myself charmed by Ra. DO pay attention to the rules – a key rule that our group misunderstood the first time around is that any player can, on his or her turn, choose to force bidding, instead of turning up another tile – not waiting for a Ra tile to turn up or the board to fill. This is critical because everyone can see who holds the trump bid in every round – if you just let the board fill up with valuable stuff, the high-bid player gets it all and leaves trash for the rest. The key insight is that if you are NOT the high-bid player, you can force a bid earlier, constantly confronting the trump-holder with the dilemma of whether to waste the winning tile on this crud or hold out for better. That way other players get bite-sized chunks and the game forces more interesting (agonizing) decisions.
I used to play Advanced Squad Leader – well, I used to study ASL, more than play it. The rules are introduced modularly, so you can always learn more, as it simulates WWII almost right down to the diarrhea.
The point at which I realized the rules were possibly too complicated:
Bicycle units were introduced. Also scouts – you can split one lone guy off a squad to scout. You can split a bike off for him too, and have a single bicyclist. There’s a completely separate rule for tanks getting hit on the front hull, where on a die roll, the driver can be stunned or blinded and the tank moves randomly one hex. On a very few maps, there are cliff faces – and there’s a rule for how to resolve combat if your tank driver is stunned and happens to go over a cliff face onto a unit below (treat it as a tank “overrun” attack without main gun or machine gun, using the hull value only).
It was when I realized that World War II had been reduced to simulating a tank falling out of the sky onto a single bicyclist – that there were rules for this – that I concluded the game was becoming a bit too Monty Python for a military simulation.
Pacific is by far the best expansion, and if not in replay-ability it certainly rivals the original in design. Japan CANNOT win with production, your military outlook is on a downward trajectory almost day one. Victory requires selling their lives for time or key territory, and at least to me is far more interesting than the ‘kill Russia then build up’ steamroller tactic of Axis in the original.
Europe is kinda lame as every game turns into two titanic clumps facing off and victory is always decided in that rolloff.
The rest are pretty awful IMHO.