Fill a large pot with 2 inches of cold water and add a generous amount of kosher salt. Bring the water to a boil. Place the live lobsters into the pot, cover with a tight-fitting lid, and steam until the shells are bright red, approximately 8 minutes. Transfer the lobsters to a baking sheet and allow them to cool until safe to handle.

Break down the lobsters by removing the claws, knuckles, and tails. Extract the meat carefully, keeping claw meat intact if possible. Cut the tail meat into bite-sized pieces. Place all meat into a bowl and refrigerate until ready to use.

Using a sturdy knife, cut the lobster shells and carapace into 1-inch pieces. Place them in a large Dutch oven or heavy-bottomed pot. Add onion, celery, garlic, basil or tarragon, cream, and crushed red pepper flakes. Bring the mixture to a boil, then reduce to a simmer. Cook uncovered for approximately 45 minutes, stirring occasionally, until the cream is reduced by half.

Strain the lobster cream through a fine-mesh sieve into a bowl, pressing firmly on the solids to extract as much liquid as possible. Discard the solids. If the liquid is more than 2 cups, it may be reduced further as needed before continuing.

Bring a large pot of salted water to a boil. Cook the bucatini or spaghetti until just al dente, stirring occasionally. Reserve 1 cup of pasta water before draining.

Return the strained lobster cream to a large skillet or Dutch oven. Warm over medium heat and season with salt as needed. Add the reserved lobster meat and butter to the sauce, stirring gently to combine.

Add the cooked pasta to the sauce and toss gently, being careful not to break apart the lobster meat. Gradually incorporate the grated Parmesan, stirring until the cheese has melted and the sauce is creamy. If the mixture is too thick, adjust by adding reserved pasta water a little at a time. Note that cream sauces thicken as they cool, so err on the side of a looser consistency when in doubt.

Transfer the pasta to a serving platter. Garnish with chopped chives, freshly ground black pepper, and additional Parmesan, if desired. Serve immediately.