1. Heat the oil in a large, wide pot over high heat until just smoking. Add the lobster shells and sear without stirring for 1 minute (the shells may stick to the pot). Stir them slightly and continue cooking, stirring from time to time, until the shells are well browned but not burned, about 5 minutes.

2. Add the brandy, garlic, shallots, and fennel. Use a wooden spoon to scrape up any bits stuck to the bottom of the pot. Stir in the tomato paste. Turn the heat to medium-low and cook until the vegetables are soft, about 3 minutes.

3. Stir in the water and bring to a boil. Boil for 15 minutes. Remove from the heat and let stand for 10 minutes. Strain through a fine-mesh sieve, pressing firmly on the solids to extract as much of the flavorful liquid as possible. Store, tightly covered, in the refrigerator for up to 3 days, or in the freezer for up to 2 months.