Bring 3 cups water to a boil in a stockpot. Place a vegetable steamer or rack in bottom of pan. Add lobster tails. Cover and steam for 8 minutes or until done. Cool to room temperature. Remove meat from cooked lobster tails; chop meat into bite-sized pieces (you should have about 1 3/4 cups chopped meat). Cover and chill.
Place lobster shells in a large zip-top plastic bag. Coarsely crush shells using a meat mallet or rolling pin. Combine shells and milk in a small saucepan over medium heat. Bring to a simmer; cook 5 minutes (do not boil). Remove from heat; cover and let stand 30 minutes. Strain mixture through a fine mesh sieve into a bowl; discard solids. Set aside.
Preheat oven to 450°.
Heat oil in a large nonstick skillet over medium-high heat. Add shallots; sauté 2 minutes or until soft. Add mushrooms; sauté 4 minutes or until liquid begins to evaporate. Stir in sherry, and cook 1 minute.
Combine reserved milk mixture, flour, black pepper, and white pepper in a small bowl; stir with a whisk until smooth. Add milk mixture to pan; bring to a boil. Cook 1 minute or until thickened, stirring constantly. Stir in basil and tarragon; cook for 1 minute, stirring occasionally. Add reserved lobster meat, cream, and salt, stirring to combine. Divide mixture evenly among each of 4 (1 1/2-cup) gratin dishes coated with cooking spray.
Combine cheese, panko, and melted butter in a small bowl; toss to combine. Sprinkle about 1/3 cup panko mixture over each gratin. Bake at 450° for 12 minutes or until topping is browned.