Add the Water (4 cup) to a large pot and season it generously with Salt (to taste). Bring it to a boil, then cook the Lobster Tail (4) until they turn a vivid red, about 5-6 minutes. Remove the tails and let them cool enough to handle. Turn off the heat temporarily.
Pour the salty lobster cooking water into a heat proof bowl or container to reserve it. Then 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 (1 dash). Add the tail shells, Fennel Bulb (1), Onion (1), and Garlic (3 clove). Let them get fragrant for a few minutes. Season with the Sweet Paprika (1/2 teaspoon) and Cayenne Pepper (1/4 teaspoon).
Stir in the Tomato Paste (1 tablespoon) and Tomato Sauce (8 ounce) and let them cook for a few minutes. Then pour in the Red Wine (2 cup) and Cognac (1/2 cup) and scrape up the bottom of the pan to deglaze.
Add the Chicken Stock (2 1/2 cup), reserved cooking water, Fresh Thyme (1 sprig), and Tarragon (1 sprig). Bring the soup to a low boil, then reduce it to a simmer and cook on medium-low for 1 hour.
Remove the tail shells and herb sprigs with a slotted spoon. Add in the lobster meat and let it warm up in the liquid for a minute.
Use an immersion blender to completely blend the soup until smooth. Stir in the Heavy Cream (3/4 cup) and Lemon Juice (1 1/2 teaspoon) and blend it a little more.
Ladle soup into bowls and drizzle a little more heavy cream on top for a pretty pattern. Sprinkle additional chopped tarragon on top if desired. Serve immediately and enjoy!