1. Cook the lobsters: Bring a large pot of salted water to a rolling boil. Add lobsters headfirst and cook for 8-10 minutes. Remove and let cool for 10-15 minutes.
2. Prepare lobster shells: Split lobsters lengthwise. Remove all meat from tails, claws, and knuckles. Clean shells thoroughly and set aside. Chop lobster meat into bite-sized pieces.
3. Make the roux: In a heavy saucepan, melt butter over medium heat. Add shallots and garlic, cooking until translucent. Sprinkle in flour and cook, stirring constantly, for 2-3 minutes until golden.
4. Add liquids: Remove pan from heat and carefully add cognac. Return to heat and cook for 1 minute. Gradually whisk in wine and fish stock, then cream. Simmer gently for 10 minutes, stirring frequently.
5. Finish the sauce: In a small bowl, whisk egg yolks. Temper by slowly adding ½ cup hot sauce to yolks while whisking. Pour mixture back into sauce, whisking constantly. Add mustard, tarragon, cayenne, salt, and pepper. Cook gently until thickened, about 3-4 minutes. Do not boil.
6. Combine and fill: Fold lobster meat into sauce. Spoon mixture into cleaned lobster shells, mounding slightly.
7. Add topping: Mix cheeses with breadcrumbs and melted butter. Sprinkle generously over filled lobster shells.
8. Broil: Place filled shells on a baking sheet. Broil 6 inches from heat for 3-5 minutes until golden brown and bubbling. Watch carefully to prevent burning.
9. Serve: Garnish with fresh parsley and serve immediately with lemon wedges.

Notes:

• For best results, bring all dairy ingredients to room temperature before starting.
• If sauce becomes too thick, thin with a little warm cream or stock.
• The filled lobsters can be prepared several hours ahead and refrigerated until ready to broil.
• Serve with crusty bread to soak up the delicious sauce.

Storage Tips:

Store leftovers in an airtight container in the refrigerator for up to 2 days. Reheat gently in a 300°F oven until warmed through. The sauce may separate slightly upon reheating – whisk gently to recombine.