Prepare the aromatic garnish: Cut the carrot into small cubes, thinly slice the shallots, and mince the garlic cloves. Place everything in a bowl. In separate small bowls, measure out 2 tablespoons flour, 2 tablespoons butter, 3 tablespoons heavy cream, and 1/2 cup white wine. Wash and chop the small tomato. Break the lobster shell into pieces and split the head in half.

Cook the aromatic garnish: In a large pot or Dutch oven (about 2.5 quarts), heat 1 tablespoon olive oil. Sauté the aromatic garnish over low heat for 10 minutes, stirring regularly, until golden.

Add the remaining ingredients: Add the lobster shell to the pot. Sauté over low to medium heat for 5 minutes. Deglaze with 1/2 cup white wine. Stir in 1 teaspoon tomato paste, 1 small tomato, 1 sprig thyme, 1 bay leaf, salt, and pepper. Mix well.

Cook: Add 6 1/3 cups water (about 1.5 liters) to the pot. Bring to a boil over medium heat, then simmer uncovered for 40 minutes.

Prepare the roux (while the bisque is cooking): In a small saucepan over low heat, melt 2 tablespoons butter, then gradually whisk in 2 tablespoons flour. Set aside in a large pot.

Strain: After cooking, strain the bisque through a fine sieve into the pot with the roux. Use a spoon to press the lobster pieces to extract all the flavor. Stir to ensure a smooth texture.

Finish: Bring the mixture to a boil over medium heat, whisking constantly. Add 3 tablespoons heavy cream. Reduce the bisque for several minutes until it reaches a velvety texture. Taste and adjust salt if needed.

Serve: Immediately pour the hot bisque into a soup bowl. Enjoy as a soup with about 10 croutons.

Variations: The bisque can also be reduced further and used as a sauce for fish or shellfish.