Bring a large pot of water to a boil and prepare the pasta according to the package instructions. Cook to Al dente. Reserve about ½ cup of pasta water to use for later.
In a skillet over medium heat, add butter, shallots, and garlic. Sauté for 1-2 minutes.
Add the cream and seasonings. Mix well and bring to a simmer. When the cream is bubbling slightly, add the grated parmesan cheese, lemon juice, lemon zest, and basil. Mix well until combined and reduce the heat to low. (do not boil sauce or the cheese will separate)
Toss in the cooked pasta with tongs. Make sure the pasta is evenly coated with the sauce. If the sauce is too thick, add a bit more reserved pasta water to reach your desired consistency. Taste and adjust with salt and pepper as needed. Then, add in the cooked lobster.
Serve and garnish with lemon zest, chopped basil or chives, and lemon wedges.