Remove lobster meat from the shell, leaving top shell intact. Remove any bits of bottom shell from the meat.
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.
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.
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.
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.
