¾ pound assorted mushrooms (baby bella, shiitake, maitake, hen of the woods, oyster) trimmed and cut into bite-sized pieced (about 4 cups)
3 cloves sliced garlic
Cover or wrap the meat and freeze it until very firm but not frozen, 45 minutes; thinly slice and set aside.
Whisk the sake, hoisin sauce, cornstarch, soy sauce, and chili flakes in a bowl; set aside,
While the meat is freezing, cook the noodles according to the package directions; drain well, cool to room temperature and make sure they’re as dry as can be (you should have about 3 cups cooked noodles. Transfer the noodles to a large bowl. Separate the scallion greens from the whites; halve the greens lengthwise, then crosswise to form 8 scallion “noodles.” Thinly slice the whites and save for the beef (below). Add the scallion greens to the noodles to the bowl along with the egg, ginger, and salt.
Heat 3 tablespoons of the oil in a 9-inch heavy skillet over medium-high heat. Add the noodle mixture, spread it out, and cook, not moving, until the underside is crispy, 5 minutes. Slide the noodle cake onto a plate, add 1 more tablespoon oil, then invert the noodle cake back into the skillet and cook until the second underside is crispy, another 4 minutes.
Remove the noodle cake from the skillet, leaving any oil in the skillet; cover with foil. Season the meat with salt, add to the skillet and cook, moving as little as possible, until browned, 2 minutes per side. Remove to a plate, add the mushrooms, raise the heat to high and cook, stirring occasionally, until tender and charred, 6 minutes. Add the remaining 2 tablespoons oil, then add the garlic and sliced chive whites and cook, stirring, until fragrant, 1 to 2 minutes. Add the sauce, bring to a boil and cook, until thickened, 3 to 4 minutes. Return the meat and mushrooms to the skillet, toss to coat, and warm through, 2 minutes. Spoon the meat onto the noodle cake. Garnish with more chopped scallions and cut the meat-topped noodle cake into quarters.