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.
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 water and 1/2 teaspoon 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 minutes. Pour through a fine-mesh strainer into a bowl; discard the solids. Set aside.
Meanwhile, melt the remaining 2 tablespoons butter in a medium skillet over medium. Add the onions, carrots, mushrooms, tomato paste, cayenne and 1/2 teaspoon salt. Cook, stirring occasionally, until the vegetables are tender, about 5 minutes. Add the wine and bring to a boil over medium-high. Reduce the heat to medium-low, add the reserved lobster stock and simmer until reduced to about 2 cups, about 10 minutes. Remove from the heat.
Puree the mixture with an immersion blender until smooth. Place over medium heat, stir in the milk and cream, and bring to a simmer. Add the lobster meat, reserving 1/4 cup for garnish.
To serve, ladle the bisque into bowls and top with the reserved lobster meat and chives.
You can make the lobster stock up to 3 days in advance.