Preparation Cook shallots and ginger in oil in a wide 5-quart heavy pot over medium heat, stirring frequently, until softened, about 5 minutes. Add tomatoes and red-pepper flakes and cook, stirring, 1 minute. Pour in carrot juice and boil, uncovered, until liquid is reduced by about half, about 10 minutes. Add mussels and cook, covered, stirring occasionally, until mussels just open wide, 4 to 6 minutes, transferring as cooked to a bowl. (Discard any mussels that remain unopened after 6 minutes.) Stir in lime juice and sprinkle with cilantro.
Preparation Cook shallots and ginger in oil in a wide 5-quart heavy pot over medium heat, stirring frequently, until softened, about 5 minutes. Add tomatoes and red-pepper flakes and cook, stirring, 1 minute. Pour in carrot juice and boil, uncovered, until liquid is reduced by about half, about 10 minutes. Add mussels and cook, covered, stirring occasionally, until mussels just open wide, 4 to 6 minutes, transferring as cooked to a bowl. (Discard any mussels that remain unopened after 6 minutes.) Stir in lime juice and sprinkle with cilantro.