Clean and chop lobster shells thoroughly to remove any bits.

Heat a splash of oil in a large stockpot over medium heat and sauté onion, carrot, and celery until fragrant and slightly softened, about 5 minutes.

Add chopped lobster shells to the pot and stir for 2–3 minutes to release flavors.

Stir in tomato paste and cook for 1 minute to deepen color.

Pour in water and add garlic, bay leaf, peppercorns, and thyme.

Increase heat to high and bring stock to a rolling boil.

Reduce heat to low and simmer gently for 2 hours, uncovered.

Strain stock through a fine sieve into a clean container and discard solids.