Prepare the Calamari Rings: Melt 1 tablespoon butter in a large sauté pan over medium heat. Setting the tentacles aside for later use, sauté the calamari rings in the butter for 2-4 minutes. Sprinkle with ½ teaspoon salt and ½ teaspoon black pepper, tossing occasionally until they reach a texture similar to al-dente pasta. Remove the cooked rings from the pan to a plate with a slotted spoon and keep warm.
Start the Soup Base: Drain any excess liquid from the pan and discard. Deglaze the sauté pan with the brandy, and melt another 1 tablespoon butter over medium heat. Add the celery, leeks, fennel, and mushrooms to the pan. Let the vegetables sweat in the pan for 5-7 minutes. This will form the base of your soup.
Deglaze the pan again with the dry sherry. Add the tomato paste, cayenne pepper, paprika, lemon juice, 1 teaspoon salt, and the bay leaf. Toss together and sauté for 3 minutes.
Stir in 3 cups of the vegetable stock and bring the mixture to a boil over medium high heat. Reduce the heat to medium low and simmer for 20 minutes to reduce the liquids. Stir in the crabmeat and cook for one minute more. In batches, carefully puree the soup in a blender and set aside.
Make a Squid Ink Roux: In the same pan, melt 3 tablespoons butter over medium high heat. Sprinkle with ¼ cup flour and quickly whisk the two together into a paste. Let the mixture cook until lightly golden. Whisking continuously, slowly incorporate the remaining 2 cups of vegetable stock, followed by the heavy cream. When the roux is creamy and completely smooth, whisk in ½ to 1 teaspoon black squid ink to give the mixture a dark grey, nearly black hue.
Finish the Seafood Bisque: Stir the pureed crab soup mixture into the squid ink roux. The soup will take on a grim grey color. Stir in the sautéed calamari rings and remove the soup from heat. Keep warm if you will be serving the soup shortly, or transfer to a container and chill until needed.