Fill a large pot with salted water and bring it to a rolling boil. Add the lobsters and cook until shells turn bright red, about 8 to 10 minutes. Remove lobsters, let cool briefly, then extract the meat and chop; keep the shells for later use.
Heat olive oil and butter in a heavy-bottomed pot over medium heat. Toss in the onion, carrot, and celery and cook, stirring frequently, until the vegetables soften, around 8 to 10 minutes. Add the minced garlic and cook just until fragrant, about 30 seconds.
Mix in the tomato paste and cook until it darkens slightly, about 2 to 3 minutes. Sprinkle the flour over the mixture and keep stirring, cooking it for another 1 to 2 minutes to form a roux.
Pour in the half cup of seafood stock combined with the lemon juice, scraping the bottom of the pot to loosen any browned bits. Allow it to simmer gently for 2 to 3 minutes.
Add the reserved lobster shells back to the pot along with the 4 cups of seafood stock, sweet paprika, bay leaf, and fresh thyme. Bring everything up to a simmer and let cook gently for 30 to 40 minutes so the flavors meld deeply.
Remove the bay leaf. Use an immersion blender to puree the broth and shells until smooth. Strain this mixture carefully through a fine sieve into a clean pot for a silky texture.
Pour in the heavy cream and fold in the chopped lobster meat. Warm the bisque gently just until heated through, about 5 to 7 minutes. Season with salt, pepper, and a pinch of cayenne if desired for a subtle kick.
Serve the bisque hot in warmed bowls and garnish if you like with extra herbs or a drizzle of cream.
