Make broth and pork belly: Preheat oven to 450°. Put pork ribs and chicken on a rimmed baking sheet. Roast, turning occasionally, until well browned, 20 to 30 minutes. Meanwhile, cook carrot, onion, and half the garlic with oil in a large pot over medium-high heat, stirring often, until browned, 10 to 15 minutes.
Add shiitakes, roasted meat, and all but 1 cup broth to pot. Discard fat from pan used for meat, add remaining cup of broth, and heat on stove, stirring, to loosen browned bits; pour into pot. Cover and bring to a boil over high heat, then reduce heat and simmer until meat is falling off the bones, about 3 hours.
Meanwhile, cut pork belly into pieces, each about 1 1/2 by 2 in. Put in a large saucepan with 3 cups water, 1 cup sake, the salt, sugar, soy sauce, ginger, lemongrass, and remaining garlic. Cover and bring to a boil over high heat, then reduce heat and simmer until very tender when pierced, 1 1/2 to 2 hours. Drain, discarding seasonings.
Preheat broiler with rack set 3 in. from heat. Put pork belly fat side up on a rimmed baking sheet. Broil until well browned, 3 to 5 minutes. Transfer to a board and slice crosswise 1/2 in. thick.
Strain broth from step 2 (discard solids), skim and discard fat. Wipe pot clean; return broth. Add kombu and heat until bubbles appear. Remove from heat; let stand 10 minutes. Discard kombu. Whisk in remaining sake and the miso. Set broth aside.
Prepare squash: Preheat oven to 425°. Toss ingredients in a 9- by 13-in. pan and bake, turning occasionally, until squash is browned and tender, 30 to 35 minutes.
Cook noodles: Heat a pot of water to boiling and cook noodles until just tender, 2 to 3 minutes. If needed, reheat miso broth until steaming; warm pork and squash in oven. Arrange a handful of bean sprouts in each of 6 large, deep bowls. Add noodles. Ladle broth over them, then arrange pork, squash, watercress, and green onions on top.
*Look for sustainably raised pork ribs and belly at stores like Whole Foods Market. You may need to special-order the pork belly from your butcher. Find kombu and jarred pickled ginger in the Asian foods aisle and miso with refrigerated foods.