First, bring a large pot of salted water to a rolling boil. Carefully add the live lobsters to the pot and cook for 8-10 minutes until bright red. Remove the lobsters, allowing them to cool slightly, then twist off the tails and claws, crack them open, and extract the meat. Chop the lobster meat into bite-sized pieces and refrigerate, reserving the shells.

Heat olive oil in a large stockpot over medium-high heat. Add the reserved lobster shells, carrots, celery, and onions, sautéing for about 10 minutes until the vegetables soften and the shells turn a deeper red. Stir in the garlic and tomato paste, cooking for another 2-3 minutes. Deglaze the pot with brandy or cognac, scraping up any browned bits from the bottom, then add the dry white wine and cook until mostly evaporated.

Pour in the water, add the thyme sprigs and bay leaves, then bring the mixture to a simmer. Reduce the heat to low, cover, and let it simmer for at least 30 minutes, or up to 1 hour, to allow the flavors to meld. Strain the stock through a fine-mesh sieve, discarding the solids, and set aside.

In a clean large pot or Dutch oven, melt the butter over medium heat. Whisk in the flour to create a roux, cooking for 2-3 minutes until lightly golden. Gradually whisk in the warm lobster stock until smooth and thickened, bringing it to a gentle simmer for 5-7 minutes. Stir in the heavy cream and return to a simmer, being careful not to boil.

Season the bisque with salt, black pepper, and cayenne pepper if desired. Gently fold in the reserved chopped lobster meat, warming it through for a few minutes. Serve immediately, garnished with fresh chopped chives.