Cook and deshell the lobster

• Bring a large pot of water to a boil and cook the lobster for a few minutes.
• Shock the lobster in an ice bath to stop the cooking.
• Separate each part and carefully deshell, keeping the pieces whole and reserving the shells.

Make the lobster broth

• Sauté the reserved lobster shells with the shallots, garlic, thyme and tomato paste.
• Pour in the chicken stock and let it simmer until rich and fragrant.
• Strain the broth and set aside.

Make the lobster butter

• Whisk a few spoonfuls of the warm broth with the butter, shichimi togarashi and lemon juice.
• Set aside for later.

Make the cheese sauce

• Use the remaining broth as a base, mixing in the heavy cream, half and half and black pepper.
• Grate the cheddar, parmesan and fontina and stir into the sauce until melted.
• Blend the sauce until silky smooth.

Assemble and bake

• Fold the al dente cavatappi into the sauce, keeping it a little saucy.
• Transfer to a baking dish and top with more parmesan.
• Broil until golden brown.

Finish and serve

• Assemble the lobster on top and brush it with the lobster butter.
• Add the head shell for presentation and serve.