9 tablespoons unsalted butter
2 green (spring) onions, including tender green parts, finely chopped
1/4 cup brandy
2 teaspoons minced fresh thyme
Salt and freshly ground white pepper
1/2 pound white button mushrooms, brushed clean and thinly sliced
1-1/2 tablespoons fresh lemon juice, or to taste
3 tablespoons all-purpose flour
1/4 teaspoon dry mustard
1-1/2 cups lobster stock, shrimp stock or fish stock
1/2 cup half-and-half
1/2 teaspoon sweet paprika
Pinch of cayenne pepper
1/2 cup grated Parmesan cheese
Preheat the oven to 425 degrees. If you have whole claws, remove the meat whole and reserve for garnishing. Cut the lobster meat into 1/2–inch dice and set aside.
In a saucepan over medium heat, melt 4 tablespoons of the butter. Add the green onions and sauté until the white part is translucent, about 3 minutes. Stir in the diced lobster and sauté until heated through, about 3 minutes. Add the brandy, bring to a simmer, and cook for 1 minute. Stir in the thyme and salt and pepper to taste. Remove from the heat and set aside.
In a nonreactive frying pan over medium heat, melt 1 tablespoon of the butter. Add the mushrooms and sauté until golden, about 5 minutes. Stir in 1 tablespoon of the lemon juice. Remove from heat and set aside.
In a nonreactive saucepan over medium-low heat, melt the remaining 4 tablespoons butter. Stir in the flour to make a roux, and cook, stirring constantly for 2–3 minutes; do not let the mixture color. Add in the dry mustard. Gradually, whisk in the stock. Cook, stirring frequently, until thickened, about 10 minutes. Stir in the half–and–half and cook, stirring frequently, for 5 minutes longer. Stir in the paprika, salt to taste, cayenne pepper, and the remaining 1/2 tablespoon lemon juice, or more to taste. Taste and adjust the seasoning.
Stir the lobster mixture and mushrooms into the white sauce. Spoon the lobster mixture into 4 individual gratin dishes (about 6 inches in diameter). Sprinkle each with one-fourth of the Parmesan. Place the dishes on a baking sheet and bake until bubbling and lightly browned, about 20 minutes. Serve at once, garnished with the claw meat, if available.