Remove the meat from the cooked lobster tails, claws, and knuckles and cut it into bite-size pieces. Refrigerate the meat while you make the bisque, and break the reserved shells into several manageable pieces with kitchen shears or a lobster cracker.

Melt 2 tablespoons (28 g) butter in a 6-quart (5.7 L) Dutch oven over medium-high heat. Add the lobster shells and cook for 6 to 7 minutes, stirring frequently, until they smell fragrant and the butter begins picking up their reddish-orange color.

Reduce the heat to medium and add the onion, carrot, celery, and garlic. Cook for 5 to 6 minutes, stirring often, until the onion looks translucent and the vegetables begin to soften without browning deeply.

Add the tomato paste and cook over medium heat for 1 minute, stirring it through the vegetables and shells. It should darken slightly and coat the vegetables before you move on.

Pour in the dry white wine and scrape the bottom of the pot with a wooden spoon. Simmer over medium-high heat for 2 to 3 minutes, until the sharp alcohol aroma softens and the wine is reduced by about half.

Add the seafood stock, bay leaf, saffron, thyme, black pepper, and kosher salt. Bring to a gentle boil over medium-high heat, then reduce to low and simmer uncovered for 25 minutes; you want small steady bubbles, not a hard boil.

Pour the broth through a fine-mesh strainer into a large heatproof bowl, pressing gently on the vegetables and shells before discarding them. Transfer the strained broth to a 3-quart (2.8 L) saucepan, add the remaining 1 tablespoon (14 g) butter, and blend carefully until smooth and slightly emulsified.

Set the saucepan over medium-low heat and stir in the heavy cream and lobster meat. Heat for 4 to 5 minutes, keeping the bisque below a boil, until steaming and the lobster reaches 165°F (74°C); stir in the lemon juice and chives, then taste and add more salt only if needed.