Peel and devein the shrimp, reserving the peels and heads and store until needed in refrigerator.

Shrimp Stock
Melt 2 tablespoons of butter in stockpot. Add all vegetable seasonings peelings and discards (onion & shallot peels, carrot root ends, tips and peels, parsley stems) to the shrimp shells and shrimp heads. Cook over high heat, stirring constantly until shrimp peels are pink (about 5 minutes.)  Flambé 1/4 cup of brandy and pour over mixture, stirring until flames die out. Add 1/4 cup of plum tomatoes and cover all with water in excess of about 2 inches (7 to 8 cups of water.) Bring shrimp stock to a boil and allow all to boil together for about 10 minutes. Strain and reserve.

Bisque
In a large saucepot, melt 6 tablespoons of butter. Add the peeled shrimp and sauté for 3 to 4 minutes until shrimp are just barely cooked. Remove shrimp with a slotted spoon, chop coarsely and reserve. Add onions, carrots and shallots to the butter that the shrimp were cooked in.  Saute together until onions are translucent. Add tarragon, salt, cayenne pepper, parsley and plum tomatoes. Cook together another 3 minutes. Flambé 1/4 cup brandy and pour over seasonings and stir until flames die out.

Sprinkle 1/4 cup flour over mixture and cook for another 4 to 5 minutes. Add 5 cups shrimp stock, 2/3 cup white wine and bring all to a boil then cover and reduce to a simmer. Simmer all together for 45 minutes. 

Puree mixture until smooth with an immersion blender. Season with Crystal hot sauce.  Stir in chopped shrimp and heavy cream and heat thoroughly - BUT DO NOT ALLOW TO BOIL.  Stir in Madeira and serve.