hoisin sauce (and sriracha, if you're so inclined), for serving
Make sure you have everything ready to go - prepped, measured, chopped, and in small bowls by the stove - before you do anything. I can't stress this enough. Once you start cooking, everything happens really quickly, so there's no time to chop something else while things are on the stove. Just saying. Put your ginger, garlic, and minced scallion in one small bowl, onion in another, carrots in yet another, and cabbage and mushrooms in a slightly larger bowl. Whisk together the soy sauce, sake, hoisin, and salt. Stir the cornstarch and water together. Get it all in an accessible place by the stove. Okay. Here we go.
Put a nonstick skillet (or wok) on the stove, turn the heat up to high, and add the oil to the pan. When the oil's hot, add the ginger, minced scallion, and garlic to the pan, and stir-fry until fragrant, around 15 seconds. Add the onion, stir-fry for 30 seconds or so, then add the carrots, and cook, tossing and stirring, for another 30 seconds. Add the cabbage and mushrooms, then dump the soy sauce mixture over everything, and stir-fry until the cabbage begins to wilt and the mushrooms start to brown, around 1 minute. Give the corn starch and water a quick stir to reincorporate, then add that to the pan, and quickly stir it in. Cook for another 30 seconds to a minute, still stirring, until the cabbage and mushrooms and both cooked through and greatly reduced in bulk. Remove from the heat, and stir in the cut scallion.
To serve, take a mandarin pancake (or tortilla), and smear a line of hoisin (and sriracha, if you want some heat) across the bottom. Spoon in some moo shu, and roll the thing up like a tiny burrito - fold the sides in over the filling, then roll it all up from the bottom. Or you can eat it like a taco (which is what I usually end up doing after my rolls fall apart). Or you can serve it over rice, but that kind of defeats the whole purpose of the moo shu experience.