A quick and easy homemade lobster stock is the base for this rich, creamy soup. Each bowl is finished with buttery, succulent lobster meat and delicate fennel fronds.
Stock: In a large pot lined with a steamer basket, bring 1 inch salted water to a boil. Steam lobster tails, top-side down, until just cooked through and tails curl under, 6 to 8 minutes for small tails and 8 to 10 minutes for large. Transfer to a plate and let cool. When cool enough to handle, use kitchen shears to cut through underside of tails. Remove meat from shells; reserve shells. Run a paring knife along back of each tail and remove vein. Cut meat into bite-size pieces; cover and refrigerate until ready to use.
Heat oil in a large pot over medium-high. Add shells and cook until fragrant and deep red in color, about 4 minutes. Add fish stock, wine, and 6 cups water; bring to a boil. Reduce heat to medium and simmer, uncovered, skimming impurities from surface, until reduced to 6 cups, 40 to 45 minutes. Strain stock into a large liquid-measuring cup or heatproof bowl. Wipe out pot. (Stock can be stored in airtight containers in the freezer up to 3 months.)
Bisque: In same pot, melt 4 tablespoons butter over medium-high heat. Add fennel, onion, and garlic and season with salt and pepper; cook, stirring occasionally, until soft and translucent, 6 to 8 minutes. Stir in tomatoes, tomato paste, thyme, bay leaf, paprika, and cayenne; cook 1 minute. Add brandy, sherry, and strained lobster stock; season generously with salt and pepper. Bring to a boil, then reduce heat to medium and simmer 30 minutes. Remove thyme and bay leaf.
Working in batches, puree soup in a blender until smooth. Return soup to pot. Let cool 15 minutes. Whisk together cream and cornstarch, then stir into bisque. Bring to a boil and cook, stirring, until thickened slightly, 1 to 2 minutes. Adjust seasoning as desired.
Melt remaining 1 tablespoon butter in a medium saute pan over medium heat. Add lobster meat and cook until just warmed through, 2 to 3 minutes. Pour soup into bowls and serve, topped with lobster, fennel fronds, and a squeeze of lemon, if desired. Soup can be made in advance and stored in an airtight container in refrigerator up to 2 days.