Bring 8 cups water to a boil in a large pot over high heat. Add the live lobster and cook until the shells turn bright red, 8β10 minutes. Transfer to a tray to cool, crack and remove the tail and claw meat, chop into bite-size pieces, and refrigerate; reserve all shells and bodies and the cooking liquid.
Pour the cooking liquid into a bowl and set aside. Return the pot to medium-high heat and add the olive oil and unsalted butter. Add the reserved shells and bodies and cook, stirring and crushing with a wooden spoon, until aromatic and deeply red, 5β7 minutes.
Add the onion, carrot, celery, and garlic; cook, stirring, until softened and lightly browned, 6β8 minutes. Stir in the tomato paste and cook until brick red and caramelized, 1β2 minutes.
Pour in the brandy and carefully ignite with a long match; let the flames subside, 15β30 seconds. Add the dry white wine and simmer until reduced by about half, 4β6 minutes.
Return the reserved cooking liquid to the pot, then add the thyme sprigs, bay leaves, and black peppercorns. Bring to a boil, add the long-grain white rice, reduce to a gentle simmer, and cook until the rice is very soft and the broth is flavorful, 30β35 minutes.
Fish out and discard the large shells, bay leaves, and thyme stems. Blend the contents with an immersion blender until as smooth as possible, then strain through a fine-mesh sieve or chinois into a clean pot, pressing on the solids to extract maximum flavor.
Stir in the heavy cream and bring just to a bare simmer over low heat for 3β5 minutes. Season with the kosher salt and black pepper. Add the reserved chopped meat to the pot and warm gently until just heated through without boiling, 2β3 minutes.
Ladle into warm bowls and garnish with chives. Serve immediately.