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1
This first part might not be pleasant, but it'll be over quickly: Working with 1 lobster at a time, place on a cutting board, belly side down, with head facing you (freezing them 8-10 minutes first will render them fairly immobile). Insert a chef's knife where the tail meets the head, and swiftly bisect head lengthwise in one fell swoop (leave tail intact). Using the back of a cleaver or a lobster cracker, crack each claw on all sides. Twist off tails and cut in half through shells lengthwise. Remove any tomalley or eggs (reserve if you like). Twist off knuckles and claws, then separate knuckles from claws using cleaver.
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2
Heat 1/3 cup oil in a wide Dutch oven or heavy pot that's large enough to hold pasta over medium-high. Season lobsters with salt and, working in separate batches and being careful not to overcrowd pot, sear lobster pieces, turning occasionally and adding more oil if pot looks dry, until shells are bright red, about 4 minutes for knuckles, claws, and tails; about 6 minutes for heads. Transfer lobsters to a rimmed baking sheet; let cool slightly. Remove pot from heat and add brandy. Return to heat and cook, scraping up browned bits, until smell of alcohol is almost gone, about 2 minutes. Transfer brandy mixture to a small bowl. Pick lobster meat from knuckles and claws; discard shells. Place in an airtight container with tails; cover and chill until ready to use. Set heads aside. Wipe out pot and reserve.
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3
Working in batches, pulse onions, carrots, and fennel separately in a food processor until finely chopped; transfer vegetables to a large bowl after each is chopped. Heat remaining 1/3 cup oil in reserved pot over medium-high. Cook vegetables, stirring occasionally, until slightly softened, 8-10 minutes. Add garlic, chiles, and 1 tsp. red pepper flakes and cook, smashing garlic with a wooden spoon and stirring occasionally, until fragrant, about 2 minutes. Stir in tomato paste, season with salt, and cook, stirring occasionally, until slightly darkened, about 3 minutes. Add wine and reserved lobster heads; bring to a simmer and cook, stirring occasionally, until reduced by one-third, about 2 minutes. Add tomato puree to pot and bring to a simmer. Reduce heat to medium and cook, stirring occasionally, until sauce is slightly thickened, 10-15 minutes. Discard lobster heads.
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4
Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until very al dente. Drain, reserving 3 cups pasta cooking liquid.
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5
Add butter, chilled lobster meat, lobster tails, pasta, reserved brandy mixture, and 2 cups pasta cooking liquid to sauce. Cook, tossing to combine and adding more pasta cooking liquid as needed, until sauce coats pasta. Because you're working with a large amount of pasta and sauce, this will take effort-toss using a long sturdy spoon in each hand, and make sure to get to the bottom of pot as you go. Add parsley and lemon juice, then transfer pasta to a platter, arranging lobster tails on top. Sprinkle with red pepper flakes and serve with lemon wedges for squeezing over.
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6
Lobsters can be prepared 6 hours ahead. Cover and chill.
