Prepare a large bowl of ice water. Set aside.

In an 8 quart or larger stockpot, add peppercorns, tarragon sprigs, 1/2 cup celery, 1/2 cup carrot, 2 cups onion, white wine, and 2 cups water. Squeeze a lemon half over pot and add as well. Cover pot and bring to a boil over high heat, reduce to a simmer, and cook for 10 minutes. Increase heat to high and add lobsters, cover, and steam for about 8 to 10 minutes, or until lobsters are bright red and their feelers are easily removed. Plunge lobsters into ice water to stop cooking. Strain steaming liquid from pot into a liquid measure. Discard solids. Once lobsters are cool enough to handle, remove meat from shells and claws, chop into 1-inch pieces, and set aside, reserving shells separately. (It is okay if the lobster is not quite cooked through at this point.)

Melt 4 tablespoons butter in a 12-inch skillet set over high heat. Add mushrooms and cook, stirring as needed, until browned, about 2 minutes. Add remaining 1/4 cup celery, 1/4 cup carrot, and 1/2 cup onion, and cook, stirring, until softened, about 3 minutes. Add 1/4 teaspoon salt, 1/4 teaspoon pepper, and a squeeze of lemon juice from remaining lemon half. Add 4 more tablespoons butter. Add lobster and heat for 1 minute, then add 1/4 teaspoon salt and 1/4 teaspoon pepper.

Melt remaining 8 tablespoons butter in a small saucepan over medium heat. Add flour and cook, stirring constantly without browning, 1 minute. Whisk in 3 cups reserved lobster-cooking liquid and reserved shells. Cook until thickened and no floury taste remains, about 10 minutes. Strain sauce through a fine mesh sieve and disregard shells. Return sauce to saucepan, stir in paprika and Cognac, and cook for 1 minute. Add chopped tarragon. Stir in cream and hot sauce and season with remaining 1/4 teaspoon salt and remaining 1/4 teaspoon pepper. Remove from heat.

Position an oven rack in the top third of oven and preheat broiler.

Divide lobster mixture and sauce among 6 gratin or broilerproof dishes. Divide Parmesan among dishes. Set gratin dishes on a baking sheet and broil until heated through, bubbly, and golden brown on top, 4 to 6 minutes. Serve immediately.

Cook's Note: This dish can be prepared up to the point where it is placed in the dishes and topped with Parmesan, then may be covered and refrigerated until ready to serve, up to 2 days. If refrigerated first, place dishes in a preheated 425-degree oven until warmed through, about 10 minutes, and then broil briefly until top is golden brown. (Time will vary depending on how long the mixture was refrigerated.)