Prep lobster and stock: Place lobster tails in a stockpot with 5 cups water and 1 teaspoon salt. Bring to a boil, lower heat and simmer 5-7 minutes, or until tails turn vibrant red and begin to curl. Using tongs, place tails in bowl to cool, reserving the hot lobster water in the pot.
Remove lobster meat from the tails and add the shells and any liquid, back into the lobster water. Simmer 20 mins while you prepare the rest of the soup. Chop the lobster meat, keeping a few pretty red pieces larger, for garnishing. Keep any red roe, which will add flavor and color. Refrigerate.
Bisque: In a heavy-bottomed pot or dutch oven, heat oil over medium-high heat. Add onion and saute 3-4 minutes until fragrant. Add garlic, celery, and thyme, and lower heat to medium, saute 5 more minutes until onions are softened. Add diced yam, cook 2 minutes, add tomato paste, paprika and bay leaf. Stir 1-2 minutes. Deglaze with the sherry, scraping up any browned bits, and cook this off, or if comfortable, carefully tilt the pot to let it ignite (stand back) and burn off the alcohol.
Strain the lobster stock. You should have about 3 cups. Pour this into the pot with the veggies. Bring to simmer on med-low, covered, and cook until the yam and celery are tender, about 15 minutes. Remove the bay leaf.
Add 1/4 the of the chopped lobster to this, and blend all until smooth and creamy (either using an immersion blender or blend in batches using a regular blender- remember, if hot, hold lid down tight with a towel.)
Pour this back into the Dutch oven. Add the cream. Taste. Season with salt and pepper. To add more depth, a tiny splash of fish sauce may help. If you like a little pep, a pinch or two of cayenne is nice.
In a medium skillet swirl the butter and smashed garlic cloves over medium-high heat until nutty and fragrant. Add the remaining lobster, and saute 1-2 minutes. Set aside a few of the biggest prettiest pieces for garnish and add the remaining smaller pieces o the soup. Simmer soup gently on low for a few more minutes. It will taste more “lobstery” as the flavors have a chance to meld. Sometimes I’ll let this sit on the stove for 1/2 hour then gently reheat before serving.
Divide among bowls, garnish the top with the bigger pieces of lobster and fresh tarragon leaves (or chives and parsley).
Serve with warm garlic bread.