1. Run fingers along the flesh side of fillets; remove any pin bones with a pair of kitchen tweezers. Turn fillets over; score skin by making 4 (1-inch) crosswise cuts in each fillet with a sharp knife.
2. Combine salt and sugar in a small bowl. Sprinkle 1 tablespoon salt mixture over skin side of each fillet; let stand 10 minutes.
3. Arrange half of cilantro sprigs in a 13 x 9-inch glass or ceramic baking dish. Arrange fillets in a single layer, skin side down, on top of cilantro. Spread remaining salt mixture over flesh side of fillets; top evenly with shallots and remaining cilantro sprigs. Cover loosely with plastic wrap. Place a cast-iron skillet or other heavy object on top of fillets to weigh them down; refrigerate 24 hours.
4. Remove skillet; set aside. Uncover fillets; drain any accumulated liquid. Carefully turn fillets over; cover loosely with plastic wrap. Place skillet or heavy object on top of fillets; refrigerate 24 hours.
5. Preheat broiler. Cut bell peppers in half lengthwise; discard seeds and membranes. Place pepper halves, skin sides up, on a foil-lined baking sheet; flatten with hand. Broil 12 minutes or until blackened. Place in a paper bag; fold to close tightly. Let stand 10 minutes. Peel. Combine bell peppers, 2 tablespoons cilantro leaves, and vinegar in a food processor; process until smooth.
6. Remove skillet from fillets; discard plastic wrap. Scrape off and discard cilantro and salt mixture; discard liquid. Using a sharp knife, remove skin from fillets. Cut fillets into 1/16-inch slices. Serve with bell pepper puree.