1. Twist off the claws and tails from the lobsters. Remove the meat, discarding the green tomalley and reserving the shells. Coarsely chop the meat to measure about 2 1/2 cups, and transfer to a bowl; cover and chill.

2. Melt 2 tablespoons of the butter in a Dutch oven over medium. Add the lobster shells, and cook, stirring often, until lightly browned, about 8 minutes. Stir in the sherry, and cook until the liquid is nearly evaporated, about 2 minutes. Add the celery, garlic, bay leaves, peppercorns, 4 cups of the water, and 1/2 teaspoon of the salt. Increase the heat to high, and bring to a boil. Reduce the heat to medium-low, and simmer until reduced to about 2 1/2 cups, about 15 to 20 minutes. Pour through a fine wire-mesh strainer into a bowl; discard the solids. Return the stock to the Dutch oven.

3. Meanwhile, melt the remaining 2 tablespoons butter in a medium skillet over medium. Add the onion, carrot, mushrooms, cayenne pepper, and the remaining 1/2 teaspoon salt. Cook, stirring occasionally, until the vegetables are tender, about 10 minutes. Add the vegetable mixture to the stock; bring to a boil over medium-high. Reduce the heat to medium-low, and simmer until reduced to about 2 cups, about 10 minutes. Remove from the heat.

4. Process the mixture with an immersion blender until smooth. Stir in the milk, cream, wine, and the lobster meat; bring to a simmer over medium. Stir together the cornstarch and the remaining 2 tablespoons water in a small bowl; add to the soup mixture, and cook, stirring constantly, until thickened, about 2 minutes. Garnish the servings with the chives.