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Fill a large pot with a tight-fitting lid with 2" of cold water. Generously season with salt and bring to a boil. Add lobsters to pot, cover, and steam until shells are bright red, about 8 minutes. (Better to slightly undercook the lobsters since you will be reheating the meat before serving.) Transfer lobsters to a large rimmed baking sheet and let sit until cool enough to handle.
Break down those lobsters! (Never picked lobster meat before? That’s what YouTube is for!) Cut or tear off small legs and eat them (they make for a great snack). Remove tail meat, cut tail meat in half lengthwise, and cut each half into bite-size pieces; transfer to a medium bowl. Remove meat from knuckles and claws, doing your best to keep the meat as intact as possible (it’s nice to have a whole claw for each person); add knuckle and claw meat to bowl with tail meat and chill until ready to serve.
With a sturdy knife, hack carapace and shells into 1" pieces and transfer to a large Dutch oven or pot. Add onion, celery, garlic, basil, cream, and red pepper flakes. Bring to a boil, then reduce heat to medium and simmer, uncovered and stirring occasionally, until cream is reduced by half, about 45 minutes.
Strain lobster cream through a fine-mesh sieve into a medium bowl, pressing on solids to get as much liquid out as possible; discard solids. (If cream isn’t reduced to about 2 cups, don’t sweat it—you can always reduce it a bit more before adding the pasta.)
Cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente. (You don’t want it to be too al dente, as it’s not going to spend that much time in the sauce.) Drain, reserving 1 cup of pasta cooking liquid.
Meanwhile, return lobster cream to Dutch oven or a wide deep skillet. Heat over medium and bring to a simmer; taste and season with salt. Add reserved lobster meat and butter to cream sauce and stir to combine.
Add pasta to Dutch oven and toss gently to combine (you don’t want to break up the lobster meat too much), adding 1 cup Parmesan a handful at a time until sauce is thick and cheese is incorporated. If sauce looks too thick, add a splash or two of reserved pasta cooking liquid to loosen it up. Remember: Cream sauces always thicken as they cool, so it’s best to err on the side of loose.
Transfer lobster pasta to a platter. Top with chives and a few cranks of black pepper. Serve with more grated Parmesan alongside.
Do Ahead: Lobster cream sauce can be made 24 hours ahead. Strain and transfer to a reasealable container and chill. Gently rewarm sauce in a Dutch oven over medium-low heat. Lobster meat can be picked 24 hours ahead. Cover and chill.