Step 1: Cook the Lobsters

Bring a large pot with a steaming rack to a boil.

Steam the lobsters for about 9 minutes until just cooked through.

Immediately transfer them to an ice bath to stop the cooking process.

Once cool enough to handle, split each lobster lengthwise.

Remove and reserve the lobster meat while keeping the shell halves intact for serving.

Reserve all shells for making the stock.

Step 2: Prepare the Lobster Stock

Heat the oil in a large saucepan over medium-high heat.

Add the sliced shallots, garlic, and celery. Cook until softened.

Add the reserved lobster shells and cook until lightly browned.

Stir in the tomato paste and cook for one minute.

Pour in the seafood stock and lemon juice mixture.

Add water and fresh tarragon.

Simmer for about 20 minutes until reduced by half.

Strain and reserve 1 cup of lobster stock for the sauce.

Step 3: Make the Thermidor Sauce

Melt the butter in a saucepan.

Cook the minced shallot until soft.

Whisk in the flour and cayenne pepper and cook for one minute.

Gradually whisk in the lobster stock until smooth.

Add the heavy cream and Dijon mustard.

Stir in the Gruyère cheese until fully melted.

Remove from the heat.

Mix in chopped tarragon.

Season with salt and pepper.

Allow the sauce to cool slightly.

Step 4: Combine the Filling

Cut the lobster meat into large bite-sized pieces.

Fold the lobster gently into the Thermidor sauce until evenly coated.

Step 5: Fill the Shells

Arrange the reserved lobster shell halves on a baking tray.

Fill each shell generously with the lobster mixture.

Sprinkle with additional Gruyère and freshly grated Parmigiano-Reggiano.

Step 6: Broil

Place under a preheated broiler for several minutes until the cheese is melted, bubbly, and golden brown.

If necessary, finish warming through in a 350°F (175°C) oven.

Serve immediately with fresh lemon wedges.