Well, it’s in the oven. I’ll let you know what happens in a couple of hours.
Thank you, unpaid research department 
Well, so far I haven’t bitten into it. Just depanned it, cooled it, and frosted it (in two layers). I’m not much of a baker, but I do bake from time to time, and these are my observations:
It’s definitely a moister type of cake. I made the recipe exactly as stated, with no substitutions or anything. The only minor difference is a 9.5"x13.5" Pyrex baking dish instead of 9"x13" pan. I baked it for 50 minutes at 350. At the end of that time, the middle looked slightly underdone, but passed the toothpick test. I was tempted to let it go another 5-10 minutes, but decided just to pull it and let it cool in the pan, outside the oven, for 10 minutes (as stated in the recipe.) When I turned it onto the baking pan, a knife ran easily around the side. I used a flexible spatula and ran it down the bottom of the pan to make sure it didn’t stick and most of it easily pulled away from the dish except for one part in the middle that was sticking to the pan. A little jiggle with the spatula and I managed to free it. It turned out onto a wire rack without a problem.
The cake itself was rather moister than most cakes and, as you observed, what I would characterize as greasy to the touch. (And, from what I can tell by similar recipes like this one, it’s supposed to be a pretty moist cake.) But I’ve made zucchini bread that had similar characteristics, so I wasn’t particularly put off by it. I let it cool for about an hour in the basement, made the frosting, and now it’s sitting in the fridge after being cut in half and made into a dual layer cake. While frosting it, I did notice that it required a bit of a light hand, as the cake itself is a bit more delicate than most. I tasted a little piece of the cake without frosting, and the cake itself tasted like a very good carrot cake. I had no issues with a sunken middle or burnt edges or anything like that. I would say the cake rose to about two to two and a quarter its batter height in the pan.
We’ll see how the taste test goes, but I may wait until tomorrow and let the cake sit in the fridge before I try it. So far, it seems fine, but I can confirm some of your observations about the cake, in terms of the moistness and greasy look, but to me that’s fine.
This is the absolute best carrot cake ever. My family has been making it for years.
I just realized that this was a misstatement. It should be, so far I haven’t cut into the finished frosted cake, but I did taste a little nibble when I was cutting the cake in half into two pieces to put on top of one another.
I’m noting that you used a Pyrex baking dish, and that earlier I said I’d had success with zucchini bread in heavy ceramic dishes. I did this cake in dark-colored nonstick cake pans … I bet that’s the difference. Just not suited for the dark, flimsy pans.
May very well be. My wife has already taken two slices of the cake, so I think it’s a success. 
My only complaint about it is that the frosting is just way too friggin sweet. Four cups of confectioner’s sugar to 1/2 cup butter and 1/2 pound cream cheese. But I wanted to do the recipe exactly as written, and my wife doesn’t seem to have any complaints. Texture of the cake itself seems right-on to me. It has retained moisture, but a cakey crumb and doesn’t, to me, really seem to have any objectionable greasiness. It really is one of the better/best carrot cakes I’ve had.
I have a great brownie recipe that ALWAYS, ALWAYS sticks to the pan. The only way I can consistently get it out of the pan in one piece (not super critical for brownies, anyway) is to cut parchment paper to size and use that in the bottom. So if the recipe otherwise tastes good, my suggestion is to use parchment paper.
+1. Parchment paper is one of the greatest things ever for baking. I use it for breadmaking when I have high hydration doughs that I want to slip into (and out of) a Dutch oven with little fuss. And the bottom browns just as well (or nearly as well–I just can’t tell the difference) as plopping the dough straight into the Dutch oven.
ETA: I should add, I discovered my most embarrassing baking mistake ever with this carrot cake. Somehow, I managed to bake it with a fork inside. I literally have no idea how it could have happened without me seeing a fork was in the dough when I poured it out into the casserole dish, but I guess the fork I was using to mix all the ingredients together full into the batter and was completely submerged when I went into the cabinet to get my baking dish out, and when I poured the batter out, somehow I didn’t notice it. I literally have no idea how that could have happened. You’d think it was pretty obvious when the batter was being poured, but somehow, I never saw it. And when I cut the cake in half to stack it into two layers, the fork must have not crossed the center, as I would have felt it there. It’s really a mystery to me. Good thing I’m not a surgeon.
gigglesnort The Carrot Cake Curse got you too, then.
Yeah, that’s worse than my salt-for-sugar substitution, which I can at least justify because I was in a foreign kitchen (in both senses, as in not my own, and not in this country) where the baking goods were kept in clear containers and the sugar was right next to the salt in a similar large amount. I poured the salt from the container only to realize as I was pouring it into my pumpkin/butternut squash pie that the granularity seemed off to me and, sure enough, when I tasted the raw filling, I realized I had poured about a cup of salt into the pie. Back to the market for more pumpkin/squash.
Meanwhile, here’s a picture of my primary taste tester with her carrot cake. Kid approved!
Cuuuute. Mine insisted we make a little coconut “nest” atop the cake/pie/whatever-it-turned-into so she could put candy eggs in it. Then she said she doesn’t like coconut, and won’t eat any.
Always count your instruments before you close.
I always make Alton Brown’s carrot cake recipe, which is nice because it’s also one layer. I generally substitute sour cream for the yogurt, though, and you can add coconut and/or raisins if you feel like it. It uses weights for most things rather than volume measure.
The OP’s carrot cake recipe is also supposed to be one layer–I just made it two because it seemed better as a four or five-inch high cake with two frosting layers as opposed to a single flat cake.
Yes, I know. I read the OP’s recipe. Some folks don’t like icing that much, and others don’t have 10 people to feed, so that’s why I mentioned the recipe I use is also one layer just in case the choice of that recipe was at least partially based on either or both of those criteria.