Preheat the oven to 325 degrees . In a large enameled cast-iron casserole, heat the vegetable oil. Season the ribs with salt and pepper and cook over moderately high heat, turning a few times, until richly browned, about 10 minutes. Transfer the ribs to a plate.

Add the sherry to the casserole and boil, stirring to scrape up the browned bits on the bottom. Add the soy sauce, mirin, onion, garlic, ginger, shiitake, brown sugar and half each of the daikon and the carrots. Return the ribs to the casserole and pour in the water. Bring to a boil. Cover and braise in the oven for about 2 1/2 hours, until the meat is very tender.

Transfer the ribs to a plate; cover and keep warm. Pick out the shiitake mushrooms, cutting off and discarding the stems. Strain the braising liquid. Return the liquid to the casserole and boil over high heat, skimming any fat from the surface, until reduced to 4 cups, about 5 minutes.

Add the remaining daikon and carrots to the casserole along with the sesame oil and shiitake mushrooms. Cover and simmer over low heat until the vegetables are tender, 20 minutes. Return the ribs to the casserole and simmer until heated through. Season with salt and pepper.

Meanwhile, bring a saucepan of water to a boil. In a bowl, cover the noodles with warm water; let stand until pliable, 5 minutes. Drain the noodles and cut into 4-inch lengths. Add the noodles to the saucepan and cook until tender but still chewy, 25 seconds. Drain and transfer the noodles to shallow bowls. Nestle the ribs on the noodles, ladle the vegetables and broth on top and serve.

The beef ribs can be prepared through Step 2 and refrigerated in the broth for up to 3 days. Skim off any fat from the surface before rewarming.