Melt butter in a sauté pan, add flour, and cook over low heat for 20 minutes. Stir frequently, not allowing it to burn. Cool and set aside this roux.
Take the live lobsters and cut the bodies in half lengthwise. Separate the tail and claws from the bodies. In two quarts of boiling water, add the tail and claws. Cook for five minutes, remove from the water and allow to cool. Reserve the cooking liquid, the lobster broth. Remove the meat from the shell and chop the lobster meat into small dice and reserve. Rough chop the raw lobster bodies.
Place a heavy bottomed ten-quart pot over high heat. Add olive oil. When the oil is smoking, add the bodies, onions, celery, carrots and garlic. Stir frequently until onions are translucent. Add cayenne, paprika, tomato paste, thyme, and pickling spice. Add brandy, sherry, reserved lobster broth, lobster base (if available) and bring to a boil and simmer for 40 to 60 minutes. Add roux to the liquid using a wire whisk and then simmer for 10 minutes.
Remove from heat and add 1½ quarts of cream, mixing thoroughly. Reserve the remainder of the cream to adjust the consistency if needed. Strain the bisque and add the diced lobster meat. Season to taste with salt and pepper as needed.