Thaw shrimp, if frozen. Peel and devein shrimp, reserving shells. Rinse shrimp; pat dry with paper towels. Coarsely chop shrimp; cover and chill until needed. Rinse shrimp shells in a colander; drain.
In a large saucepan combine shrimp shells and broth. Bring to boiling; reduce heat. Simmer, uncovered, for 5 minutes. Strain broth into a bowl; discard shells. Rinse out saucepan. In the saucepan cook soba according to package directions; drain. Return to hot pan; cover and keep warm.
Meanwhile, in a Dutch oven cook mushrooms, green onions, ginger, and garlic in hot peanut oil over medium heat about 5 minutes or until mushrooms are tender. Stir in strained broth, bok choy, sweet potato, and water chestnuts. Bring to boiling; reduce heat. Simmer, uncovered, about 6 minutes or until bok choy and sweet potato are tender. Stir in shrimp and, if desired, sesame oil. Cook about 1 minute more or until shrimp are opaque.
Serve soup in bowls over soba. Serve with lime wedges and, if desired, chili sauce, cilantro, and sesame seeds.