Note: To roast garlic, cut the top off to reveal the tops of the cloves. Drizzle the top with olive oil and a sprinkle of salt, then wrap it in foil. Roast at 350ºF for 1 hour, then refrigerate it and use it as needed! It keeps for a long time.
Add water to a large pot and salt generously. Bring it to a boil, then boil lobster tails until they turn a vivid red, about 5–6 minutes. Remove tails and let cool enough to handle. Turn off heat temporarily. Pour thesalty lobster cooking water into a heat proof bowl or container to reserve it. Crack the tails open and remove the meat, trying not to break the tails too much into little pieces. Chop up the meat into tiny pieces and store it in the refrigerator while the soup cooks.
Put the pot back onto medium high heat and heat up the olive oil in it. Add the tail shells, fennel, onion and roasted garlic and let them get fragrant for a few minutes. Season the mixture with the paprika and cayenne. Then stir in the tomato paste and tomato sauce and let them cook for a minutes. Pour in the wine and cognac and scrape up the bottom of the pan as they help deglaze it. Finally, add the chicken stock, reserved cooking water, thyme sprig and tarragon sprig. Bring the soup to a low boil, then reduce it to a simmer on medium low for 1 hour.
When the hour is up, remove the tail shells and herb sprigs with a slotted spoon. Add in the lobster meat and let it just warm up in the liquid for a minute. Then use an immersion blender to completely blend the soup until it is smooth. Stir in the heavy cream and lemon juice and blend it a little more. Then just ladle it into bowls and drizzle a little more heavy cream into it for a pretty pattern and sprinkle additional chopped tarragon on top. Serve immediately and enjoy!