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 shells (save shells); chop meat into bite-sized pieces.
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 1/4 of the panko mixture over each gratin.