Preheat the oven to 425°F 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 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 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 saucepan over medium heat, melt the remaining 4 tablespoons of butter. Stir in the flour to make a roux and cook, stirring constantly, for 2-3 minutes; do not let the mixture color. 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, Tabasco sauce to taste, and the remaining 1/2 tablespoon of lemon juice, or more to taste. Taste and adjust seasoning.
Stir the lobster mixture and mushrooms into the white sauce. Spoon the lobster mixture into 4 individual gratin dishes. 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.