Step 1
Cut bottom shell of lobster tails using kitchen shears along each side where it meets the top shell; do not remove the shell at this point.
Step 2
Melt 2 tablespoons butter in a skillet over medium-high heat. Place lobster tails cut side down in the butter. Cook covered about 3 minutes. Turn lobster tails over (they are curved so will lean to 1 side); cover and cook 1 minute. Lean them to the other side; cover and cook 1 minute more.
Step 3
Remove from heat. Transfer lobster tails to a bowl. Remove any shell pieces from the skillet.
Step 4
Place 1 tablespoon butter in the skillet over medium heat; add shallots and pinch of salt. Cook and stir until pan starts to deglaze and shallots soften 3 to 4 minutes. Add mushrooms; cook stir until golden brown 5 to 6 minutes. Add flour; stir until mushrooms are coated and cook for 2 minutes. Pour in cognac; use a kitchen torch to flame cognac to burn off alcohol. This will take just a few seconds. Add milk; cook and stir until pan is completely deglazed and milk begins to thicken and turn bubbly about 2 minutes. Remove from heat. Stir in creme fraiche. Let cool.
Step 5
Remove lobster meat from the shell leaving top shell intact. Remove any bits of bottom shell from the meat.
Step 6
Preheat oven to 450 degrees F (230 degrees C). Line a baking pan with crinkled up foil. Make 4 channel-like indentations in the foil to hold and stabilize the lobster tails as they bake.
Step 7
Slice lobster meat down the middle and check for and remove any dark veins. Chop into 1/2-inch pieces. Transfer meat to a mixing bowl. Add cooled sauce tarragon pinch of salt and cayenne pepper. Stir gently until well blended.
Step 8
Divide mixture among the lobster tail shells; place in prepared foil-lined pan. Sprinkle each lobster tail with about 1/2 teaspoon bread crumbs and 1/2 teaspoon grated cheese. Drizzle melted butter on the stuffed shells.
Step 9
Bake in preheated oven until nicely browned and heated through about 10 minutes. If you like you can turn on the broiler and broil 1 minute until tops are golden and crispy.