How to Make It
Bring a large pot of water to a boil. Add the lobsters, head first, and cook for 5 minutes. The water may not come back to a boil, but the lobsters will be almost cooked through. Using tongs, transfer the lobsters to a large rimmed baking sheet to cool. Reserve 3 cups of the cooking liquid.
Crack the claws and knuckles and remove the meat. Using scissors, cut along the inside of the tail shells and remove the meat. Remove and discard the dark intestinal veins from the tails. Put all of the lobster meat in a bowl and refrigerate. Remove the dark-green eggs from the bodies and top of the tails and reserve. Using poultry shears, cut the shells into 2- to 3-inch pieces.
In a large saucepan, heat the vegetable oil. Add the lobster shells and cook over high heat, stirring, until they are sizzling and just starting to brown, about 5 minutes. Add the shallots and thyme and cook over moderately high heat until the shallots soften, about 4 minutes. Add the tomato and cook until the juices evaporate. Add the wine and boil until reduced by half, about 4 minutes. Add the reserved 3 cups of lobster cooking liquid and bring to a boil. Simmer over moderately low heat until reduced to 3 cups, about 20 minutes. Strain the lobster broth through a fine sieve and return it to the saucepan.
Bring the lobster broth to a simmer over moderately high heat. Put the reserved lobster eggs in a small sieve that fits in the pan of lobster broth and simmer until they turn bright red, about 1 minute. Remove the sieve from the broth and press the eggs through the mesh into a small bowl to separate them.
Return the lobster broth to a boil, add the heavy cream and boil over high heat until reduced to 1 cup, about 7 minutes. Reduce the heat to low. Slice the lobster tails 1/3 inch thick.
In a large pot of boiling salted water, cook the pappardelle, stirring, until al dente. Drain the pasta and return it to the pot. Add 1 tablespoon of the butter, toss well and cover.
Add the lobster meat and eggs to the sauce and bring it just to a simmer. Season with salt and pepper. Remove the pan from the heat and whisk in the remaining 4 tablespoons of butter, 1 tablespoon at a time, occasionally moving the pan back to the burner to keep the sauce very warm. Stir in the chives. Pour the sauce over the pappardelle and toss to coat. Transfer to shallow bowls and serve.