Source: Based Cooking (community recipes)
Kill the lobster cleanly with a sharp knife through the head, then split it lengthways. Scoop out the grey-green tomalley (liver) and any coral (roe) — reserve both for enrichment later. Extract the meat from the claws, knuckles, and tail; dice it into bite-sized pieces and refrigerate. Smash the shells and carcass into rough fragments.
Heat olive oil in a stockpot over medium-high heat and toast the shellfish shells for 5–6 minutes, stirring occasionally, until they lose their translucence and smell sweet and briny. This classical-french-technique of shell-roasting extracts the carapace's umami compounds without burning them. Pour off the oil, add the butter, then introduce your aromatic-vegetables|aromatics: the diced onions, carrots, and shallot. Cook for 3 minutes until they soften and begin to catch. Add the garlic, stir for 30 seconds, then pour in a small glass of cognac and flambé — this burns off the ethanol and deepens the spirit's sweetness. Lower the heat to medium.
Add the tomato purée and fresh tomatoes, cooking for 2 minutes to caramelise the tomato sugars slightly. Dust the flour over the shells and vegetables, stirring well for 1 minute — this roux will thicken the finished soup-making|bisque and bind the fat-soluble flavours into the liquid. Pour in the white wine, scraping the pan bottom to lift all caramelised fragments; the acid denatures proteins and releases flavour from the shells. Once the wine has reduced by half, pour in the fish stock. Tie the bay leaf and thyme sprigs in muslin and submerge them.
Simmer gently for 30 minutes — the slow extraction softens the shell structure and allows collagen to migrate into the liquid, building body. Pass the entire pot through a fine sieve, pressing gently on the solids to extract every bit of flavoured liquid without clouding the bisque with shell fragments. Return the strained broth to the pot and bring to a light simmer. Stir in the reserved diced lobster meat and cook for 3 minutes — no longer, or it toughens. Press the reserved tomalley and any coral through a fine sieve into the pot off the heat; this adds richness and a subtle iodine note. Finish with a pour of heavy cream, season sharply with salt and white pepper, and ladle into warm bowls.
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