I like to sautee my green BEANS with butter and GARLIC.
Ok, this recipe, minus the chicken broth and minus the water and plus some mushrooms. Sautee the green beans and mushrooms in the half cooked onions in the grease. A cast iron skillet works best for this, or something else good and heavy. You’re going to be cooking these a while, but they don’t need constant attention, so start 'em first and stick 'em on the back burner over medium to medium low heat and give 'em a stir once in while as you cook the rest of your dinner. About 45 minutes or so, and someone will wander into the kitchen and warn you you’re burning the green beans. Whack 'em with a spoon and tell them they can cook dinner next time or gtfo. Another 5 minutes and they’ll be done. Add some slivered almonds in the last 2 minutes if your mother in law or the Queen is coming to dinner.
You want them nice and cooked, nice and browned and caramelized - as **pulykamell **says, many people will tell you they’re overcooked. Until they try them, and then they won’t say another word, because they’ll be stuffing their maw with the best damn green beans ever created.
This actually works really well with canned green beans and canned mushrooms, even. Although fresh is better. French style is best, but regular cut works too. And start with twice as many as you think you need, because it cooks down a lot.
I also like frozen green beans steamed in the microwave until just warmed through and then drizzled with balsamic vinegar.
If I can find them very fresh I’ll cook them slowly in a little tomato, oregano, lemon juice, olive oil, and garlic. Goes very nicely with roasted lamb and potatoes.
I suppose I should probably offer up a concrete recipe, too. 
This recipe for Turkish style green beans is one of my favorites. I guess that one’s only a 45-minute braise followed by a 45-minute cool down, so not quite the “cooked at least an hour.” But what originally turned me on to slow-cooked green beans was this Washington Post article on green beans braised for three hours. I didn’t really think I much enjoyed green beans (having been pummeled by the mantra of "do not overcook green beans, just lightly saute/steam them) until I tried that recipe.
Those beans do look really good, but I can definitely see myself making the leeks very soon.
I like them cooked for a while. They’re never mushy, just very, very tender. I like them cooked in a stock I make from leftover country ham scraps.
So now I have to explain to my wife that tomorrow I want green beans for dinner… prepared 23 different ways. (we need a drooling smilie).
I should clarify that I like green beans in any form, but I still think the best flavor comes out after they have been simmered for at least an hour with bacon, onion, and a bit of red pepper. My wife disagrees and wants that crispy texture, but I think the flavor they get during the simmer is well worth the less than crispy texture.
I like 'em Asian style! Saute 1/2 pound of beans in olive oil for about 5 minutes, then add garlic and about 1/4 cup of low sodium soy sauce. Stir fry until tender-crisp and finish with a teaspoon or so of dark sesame oil. Yum, yum, yum!
That reminds me - a local southern-style restaurant does deep-fried battered green beans. They’re delicious! So I’m seconding the tempura suggestion.
Another salad variant: lightly cooked, drained, and while still warm mixed with olive oil, vinegar, minced onion or scallion, grated parmesan cheese, and herbs of your choice (Provencal mix is good).
They’ll keep well this way for a day or so. At serving time, add salt to taste and sliced tomatoes.
Roasted green beans are great. 425 degrees, toss in some oil and salt, bake for about 10-15 min.
In a pressure cooker with bacon and onion. Very fast, and they come out nice and crisp. And there’s bacon.
Seems to be getting kind of hot in here…
Honestly, the only way I ever really eat them is reheated from a can or frozen I haven’t eaten them fresh since my grandmother stopped having a garden. I don’t think I’ve seen fresh for sale.
With canned, for sure, there’s really not much choice. We throw in occasional bits of ham and some olive oil, but we definitely just simmer them. My mom always said it tasted better on the stove, but I’ve never really detected a difference using the microwave. If you boil 'em, you’ve gone way too far.
Sichuanese dry-fried was my first thought as well. I do it to Fuchsia Dunlop’s recipe from Every Grain of Rice
This one. Optionally with a little Maggi Hot and/or tossed with nutritional yeast.
Saute some onions and lots of bacon, add tomato sauce or crushed tomatoes, lots of red pepper flakes (or the chili pepper of your choice), a little vegemite, stir up, add green beans, cook in a pressure cooker for about 12 minutes on high, and voila!
If I can’t get fresh ones I’ll buy them frozen. I usually cook them for about 20mins, so they’re not mushy but soft. I serve them with some white sauce and dill and a side of patatoes - they’re my favorite simple meal because they remind me of my great grandmother
Stick the beans in the microwave with a splash of water. At the same time start melting some butter/coconut oil in a frying pan on a low heat.
Microwave the beans for a while. As soon as you smell beans or the butter/coconut oil is ready turn them off.
Turn up the heat under the frying pan flat out and chuck in the beans. Grind fresh black pepper over them, zest a citrus fruit over them and chuck in some crushed/flaked nuts. Almond meal works OK too.
I fry up some cut up bacon with some chopped onions. While that’s cooking I boil the green beans until they’re done. When the bacon is crispy and the onion is transluscent, I add some vinegar and some sugar and stir until combined, then add the drained green beans to it.
Basically a wilted lettuce salad without the lettuce and with green beans.