Fill a large pot with the water and 1 teaspoon of the sea salt.
Bring the water to a boil.
Add the lobster tails and cover the pot with a lid.
Let the tails boil until they are bright red, about 5 minutes.
Remove the lobster tails, reserving the cooking liquid.
When the lobsters have cooled slightly, remove the meat from the shells, reserving the meat and any liquid that comes out of the shells.
Return the lobster shells to the pot of water and bring the mixture to a boil.
Reduce the heat to medium-low and let it simmer until the flavor of the lobsters are drawn as much as possible from the shells, about 15 minutes.
Chop the lobster meat into bite-size pieces and refrigerate.
Heat 2 tablespoons of the butter and 1 tablespoon of the olive oil in a large, heavy-bottom pot over medium heat.
Add the onions, the carrots, the celery, the thyme, and 1 teaspoon of the tarragon and cook until soft, about 5 minutes.
Season the veggie mixture with the bouillon powder, the salt, the black pepper, and 1/2 teaspoon of the cayenne pepper.
Stir 4 cloves of the minced garlic into the veggie mixture and cook until fragrant, about 1 minute.
Mix the tomato paste into the veggie mixture and cook, stirring to coat the veggies, until fragrant, about 1 minute.
Sprinkle the flour over the veggie mixture and cook, stirring occasionally, until the raw flour smell disappears, about 2 minutes.
Pour the wine into the veggie mixture and allow it to simmer until reduced by half.
Stir 4 cups of the homemade lobster stock into the veggie mixture.
Reduce the heat and gently simmer, stirring occasionally, until the liquid has thickened slightly and the flavors have blended together, about 30 minutes.
Take the soup mixture off the heat and transfer it to a blender.
Blend the soup mixture until smooth.
Return the soup mixture to medium-low heat and stir in the heavy cream.
Melt the remaining butter in a skillet pan over medium heat.
Add the remaining garlic and sauté until fragrant, about 30 seconds.
Add the chopped lobster meat to the garlic mixture and season it with the salt, the pepper, and the remaining cayenne.
Lightly sauté the lobster, stirring occasionally, until just warmed through, about 1 minute.
Mix 3/4 of the lobster meat into the bisque.
Pour the bisque into serving bowls and top each bowl with an even amount of the remaining lobster meat.
Sprinkle on the extra tarragon and serve.