Place large, heavy-bottom pot on stovetop over medium-high heat. When pot is warm, add 2 tablespoons olive oil. Continue heating pot, swirling and tilting pot as needed to distribute oil across bottom of pot, until oil is hot and shimmery.
When oil is hot, add lobster shell pieces to pot. Sauté shells, stirring occasionally, approximately 5 minutes or until shells have darkened. Be careful not to burn shells.
Once shells have darkened, add slices of 1 medium onion, ½ cup sliced celery, ½ cup sliced carrots, 1 head garlic, ¾ cup chopped red tomato, 2 sprigs fresh tarragon, 6 sprigs fresh thyme, 2 large bay leaves, and ½ teaspoon whole black peppercorns to pot. Stir to incorporate.
Add ½ cup brandy and ½ cup sherry to pot and stir to incorporate. Bring mixture to gentle boil, then allow mixture to boil 5 minutes or until majority of liquid has evaporated.
Add 2 cups bottled clam juice, 2 cups cooking liquid (reserved after boiling lobster tails), and reserved lobster juices to pot. Stir to incorporate, then reduce heat under pot to medium.
Allow mixture to come to strong simmer. Simmer mixture 15 minutes.
After 15 minutes, remove pot from heat and replace with large saucepan. Fit large sieve into saucepan, then carefully pour contents of pot into sieve.
Strain soup, firmly pressing on solid ingredients with back of wooden spoon to express as much liquid from ingredients as possible. Once strained, discard leftover solid ingredients.
Add ¼ cup tomato paste to saucepan and whisk until tomato paste is fully incorporated.
Simmer soup over medium heat approximately 15 minutes or until soup has reduced to approximately 3 cups.
Once soup has reduced, remove saucepan from stovetop and let cool slightly.
Optional: If making lobster bisque in advance, stop here. Transfer soup to airtight container and let cool completely, then refrigerate up to 1 day. 20 minutes prior to serving, reheat soup on stovetop over low heat, then continue with instructions below.
Add ½ cup whipping cream (at room temperature) to soup and whisk to incorporate, then return saucepan to stovetop. Simmer mixture 5 minutes.
While mixture simmers, add 1 tablespoon cornstarch and 1 tablespoon cold water to small mixing bowl. Whisk until ingredients are fully combined.
Add prepared cornstarch slurry to saucepan and stir to incorporate. Increase heat under saucepan to medium-high and bring mixture to boil, then let mixture boil 2 minutes or until slightly thickened.
Once soup has thickened, taste bisque and add ½ teaspoon salt and freshly cracked black pepper to taste. Stir to incorporate ingredients.
Remove saucepan from heat and let cool slightly, then add chopped lobster meat to soup. Stir to incorporate, then return saucepan to heat.
Simmer mixture, stirring constantly, until lobster meat is just warmed-through.