Beef, Pumpkin, and Shiitake Soup

Combine beef and 10 cups water in a pot. Bring to a boil, skimming any foam and impurities from surface. Add half of onion, 2 teaspoons salt, peppercorns, bay leaf, thyme, pumpkin peels, shiitake stems, and cabbage leaf. Gently boil over medium-low heat, partially covered, until meat is falling-off-the-bone tender, about 2 hours. Strain (if you want a clear broth, line a fine-mesh sieve with cheesecloth), reserving broth and beef and discarding remaining solids. Wipe pot clean.

Return broth and beef to pot. Coarsely chop remaining onion half and transfer to pot, along with pumpkin and mushroom caps. Simmer over low heat until pumpkin is crisp-tender but still holding its shape, about 10 minutes. Stir in cabbage and basil sprig; simmer until cabbage is tender but still vibrant green, about 5 minutes more. Remove and discard basil. Stir in fish sauce and lime juice. Add more salt, fish sauce, and lime to taste, and serve topped with basil leaves.