Some research on the Serious Eats boards led me to believe (I'm not 100% sure this is correct, so feel free to chime in below in the comments) that the mutant vegetable was a kabocha, also referred to as a Japanese pumpkin. Bittman's advice in How to Cook Everything is that even though winter squashes are all different, they're interchangeable in recipes--they may need a bit more or less cooking time or liquid as they cook, but you can feel free to mix and match. So that's what I did: I took the recipe for braised and glazed winter squash with soy variation, and used this enormous green pumpkin instead of the usual butternut, which Bittman warned me would be better because it's "easier to deal with than the others."
That's probably true. Peeling this thing was a pain in the ass, and I tried roasting its seeds thinking they'd be just like pumpkin seeds. They were not. They tasted weird and cooked really unevenly, so some ended up soggy and others were burnt to a crisp.
For the squash, you cut the vegetable up into cubes, then heat some oil in a pan with garlic. After the garlic starts to cook, you add the squash, some water, and some soy sauce and salt and pepper. Let this simmer, covered, until the squash is tender. Bittman says this takes about 20 minutes, but with the green pumpkin, which I suppose is much more dense than butternut squash, it took over an hour and still wasn't as tender as I would have liked. It was pretty tasty though.
My camera died before dinner was ready, so there's no photo of the finished product this time; instead, I give you this picture of Lily playing with a pumpkin.