If using live lobsters, bring a large pot of salted water to a boil. Carefully add the lobsters and cook for about 8-10 minutes until bright red. Remove them from the pot and let them cool. Once cool, remove the lobster meat from the shells and chop it into bite-sized pieces. If using pre-cooked lobster, simply chop it into bite-sized pieces.
In the same pot, cook the linguine according to package instructions until al dente. Reserve about 1 cup of pasta water, then drain the pasta and set aside.
In a large skillet over medium heat, melt the butter. Add the minced garlic and chopped shallot, and sauté for about 2-3 minutes until fragrant and softened.
Pour in the white wine and let it simmer for about 3-4 minutes until reduced by half.
Add the heavy cream, cherry tomatoes, and lobster meat. Stir gently to combine and simmer for about 5-7 minutes, allowing the sauce to thicken slightly.
Mix in the cooked pasta, adding reserved pasta water a little at a time until reaching your desired sauce consistency.
Stir in the grated Parmesan cheese, and season with salt and pepper to taste. Remove from heat and fold in the fresh basil.
Serve immediately, garnished with lemon wedges.