Using kitchen shears, cut along length of each lobster tail shell. Remove vein running through tail. (If needed, rinse the lobster tails under cold running water. Drain and pat dry with paper towels.) Using your fingers, loosen the meat from the shells; remove and set aside. Cut or break the shells into large pieces.

Melt butter in a large saucepan over medium heat. Add lobster shells, onion, celery, garlic, bay leaves and thyme; cook, stirring often, until the vegetables are softened, about 10 minutes. Sprinkle flour over the vegetable mixture; cook for 2 minutes, stirring constantly. Stir in tomato paste until the mixture is well coated. Add brandy; cook, stirring constantly, until mostly evaporated. Pour in seafood (or lobster) stock and chicken broth and bring to a boil over high heat. Reduce heat to low and simmer, stirring occasionally, until just slightly thickened and flavors have melded, about 30 minutes.

Working in 2 batches, transfer the mixture to a blender. Remove center piece of blender lid (to allow steam to escape); secure the lid on the blender and place a clean towel over the opening. Process until blended, about 2 minutes. (Use caution when blending hot liquids.) Pour through a fine-mesh sieve into a large measuring cup, pressing with the rounded side of a ladle to squeeze out the liquid; discard solids. Strain the batch a second time. Repeat with the remaining soup: blend, then strain twice.

Stir cream, vinegar, cayenne, salt and 1 tablespoon lemon juice into the bisque. Divide evenly among 8 warm bowls. Toss the reserved lobster meat with the remaining 1/2 tablespoon lemon juice. Top each serving with some of the lobster, sprinkle with chives and serve immediately.