Lobster Thermidor is a French dish made with fresh tender lobster in a cream sauce, baked in lobster shells and topped with gruyere cheese.
Add 8 cups water to a very large stockpot. Bring to a boil over high heat. Add lobsters and cover. Adjust heat as needed to prevent water from boiling over. Cook for 11 minutes. Remove lobsters from pot to cool.
While lobsters are cooking, combine clam juice, wine, and cognac in a saucepan. Bring to a boil and boil until reduced to ¾ cup, about 20 minutes.
Meanwhile, for the reduced clam juice and wine, turn heat down to medium. Add heavy cream and simmer for 5 minutes, stirring often. Remove from heat. Season to taste with salt and pepper. Add ½ cup shredded gruyere cheese and stir until cheese is melted and sauce is smooth.