Lobster Thermidor

Ingredients

• 2 lobsters (about 2 ½-pounds each)
• 2 cups white wine
• 2 whole garlic cloves
• 1 large carrot (rinsed and cut into one-inch pieces)
• 1 large yellow onion (peeled and quartered)
• 1 large celery stalk (cut into one-inch pieces)
• 1 quart water
• 5 black peppercorns
• 8 thyme sprigs
• 1 small bunch Italian flat-leaf parsley
• 3 large tarragon sprigs
• 2 bay leaves
• 8 ounces button mushrooms (sliced)
• 2 large whole button mushrooms
• 1 stick butter (divided (8 tablespoons))
• ½ cup shallots (minced)
• 1/3 cup cognac or brandy
• 5 tablespoons all-purpose flour
• 1 tablespoon Dijon mustard
• 2 tablespoons grated Parmesan cheese
• Pinch cayenne pepper
• 1 ½ cups light cream or half and half
• Kosher salt and white pepper (to taste)
• ½ cup Gruyere cheese (shredded)
• 2 tablespoons fresh flat-leaf Italian parsley (chopped)