In a large pot, bring salted water to a rolling boil. Carefully add the live lobsters and cook for 8-10 minutes until they turn a vibrant bright red and the shells are firm. Using tongs, remove the lobsters and plunge them into an ice bath to cool rapidly, stopping the cooking process. Once cool, crack open the shells and remove the meat, being careful to keep the shells intact.

In a large skillet over medium heat, melt 4 tablespoons of butter until frothy. Add finely chopped shallots and sauté for about 2 minutes until they become translucent and fragrant. Then, add sliced mushrooms and sauté until they are tender and their moisture has evaporated, about 3-4 minutes.

Pour in ½ cup of white wine, raising the heat to medium-high, and let it simmer for about 3-4 minutes until reduced by half. Then, reduce the heat to medium-low and slowly stir in 1 cup of heavy cream. Allow the mixture to thicken for approximately 3 minutes while gently stirring.

Chop the lobster meat into bite-sized pieces, then gently fold it into the creamy sauce. Add 1 teaspoon of paprika and half of the grated Parmesan cheese, stirring until evenly incorporated. Season the mixture with salt and pepper to taste.

Preheat your broiler to high. Carefully spoon the lobster mixture back into the reserved lobster shells, generously mounding it. Top each filled shell with the remaining grated Parmesan cheese. Place the filled shells on a baking sheet and broil for 3-5 minutes, or until the cheese is golden brown and bubbly.

Remove the French Lobster Thermidor from the oven and let it cool for a minute before serving. Present it hot, ideally accompanied by a light Caesar salad or crusty bread.