Step 1

Place a rack in lower third of oven; preheat to 325°. Heat 1 tablespoon oil in a large heavy pot over medium-high heat. Season short ribs lightly with salt and pepper. Working in batches, cook short ribs, reducing heat as needed to avoid overbrowning, until brown on all sides, 10-12 minutes per batch. Transfer short ribs to a plate; drain all but 1 tablespoon drippings from pot.

Step 2

Add shiitake stems, scallion whites, and ginger to pot. (Cover and chill remaining mushrooms and scallions.) Cook, stirring often, until fragrant, about 2 minutes. Add 2 cups broth, soy sauce, sake, brown sugar, star anise, and cinnamon; bring to a simmer.

Step 3

Return short ribs to pot, nestling them bone side up in liquid in a single layer. Cover pot, transfer to oven, and braise until ribs are fork-tender, about 2 1/2 hours.

Step 4

Using tongs, transfer ribs to a shallow baking dish bone side down. Strain liquid from pot through a fine-mesh sieve into a large measuring cup; discard solids. Skim fat from cooking liquid. Add broth as needed so liquid measures 1 1/4 cups and pour over ribs. Let cool completely, then cover with foil and chill for at least 3 hours. DO AHEAD: Ribs can be cooked 2 days ahead. Keep chilled.

Step 5

Preheat oven to 350°. Bake short ribs, covered, until heated though, about 25 minutes. Remove 1/2 cup liquid from baking dish; set aside.

Step 6

Heat remaining 2 tablespoons oil in a large skillet over medium-high heat and cook shiitake caps until brown and tender, about 3 minutes. Add reserved braising liquid and all but 1 tablespoon scallion greens and season with salt.

Step 7

Place cooked noodles on a large, deep platter. Arrange short ribs on top of noodles; drizzle sauce from dish over. Spoon mushrooms with sauce over. Sprinkle with remaining scallion greens.