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To prepare the lobsters: Use a large chef s knife.
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If you wish, stun the lobsters a bit by putting them in the freezer for 15 minutes.
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Put the tip of your chef s knife on the lobsters head, about 2 inches or so back from the eyes.
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Push the knife straight down, then through to split between the eyes.
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Hold the lobster with a towel where the claws meet the body, and twist to remove the claws.
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Twist or break the claws from the knuckles, and crack both with the back of the knife to make it easier to open when serving.
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Twist walking legs off the body.
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Split the lobster body and tail in half lengthwise, clean body cavity, leaving in the tomalley (the green digestive part of the lobster).
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Cut the tail from body.
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Bring a large pot of salted water to boil for pasta.
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After you have simmered the sauce with the lobster for 10 minutes, slip the spaghetti into the boiling water, and cook until al dente.
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Spread the flour on a rimmed baking sheet.
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Dredge the cut-side pieces of lobster body and tail (but not the claw pieces or walking legs) in the flour, tapping off the excess.
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Pour the vegetable oil into a large Dutch oven, and set over medium-high heat.
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Slip the body and tail meat into the pot, cut side down, and cook just to seal the meat, about a minute or so.
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Remove the pieces to a plate.
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Add the claw pieces, and cook just until they begin to change color, about a minute.
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Remove the claws to the plate.
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Pour off the vegetable oil, return the pot to medium heat, and pour in 3 tablespoons of the olive oil.
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Add the onions and garlic.
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Saute a few minutes, then add 1/2 cup of the pasta cooking water, and simmer to soften the onions, another 2 to 3 minutes.
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Increase the heat to let the water boil away, and clear a space in the pan to make a clear dry spot.
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Plop in the tomato paste, let sizzle a minute or two, then stir the tomato paste into the onions.
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Add the crushed tomatoes, and slosh out the cans with 2 cups of the pasta cooking water, adding those to the pot as well.
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Bring the sauce to a rapid boil, and stir in the peperoncino and salt.
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Add all of the lobster except for the tail pieces, and let simmer until the sauce is thickened, about 10 minutes.
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Add the tail pieces, and simmer until the meat is just cooked through, about 2 to 3 minutes more.
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When the sauce is ready, transfer about half of the sauce (without the lobster) to a large skillet, and bring to a simmer; add the cooked and drained pasta.
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Drizzle with the remaining tablespoon of olive oil, and sprinkle with the shredded basil.
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Toss to coat the pasta with the sauce.
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Serve the pasta in shallow bowls, with the extra sauce and the lobster-tail pieces over top.