Place chile on a foil-lined baking sheet; broil 3 inches from heat for 8 minutes or until blackened and charred, turning after 6 minutes.
Place in a paper bag; fold to close tightly.
Let stand 10 minutes. Peel, and discard skins. Chop chile.
Combine chopped chile, wine, shallots, 3 tablespoons cilantro, salt, and next 4 ingredients (through clam juice) in a Dutch oven; bring to a boil. Cover, reduce heat, and simmer 8 minutes.
Add mussels; cover and cook 3 minutes or until shells open.
Remove mussels from pan with a slotted spoon. Discard unopened shells. Keep warm.
Bring wine mixture to a boil over high heat; cook until mixture is reduced to 1 cup. Stir in lime juice and butter.