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Bring a large pot of water to a boil, and salt generously. Add the pasta, and boil according to package directions until al dente, tender but still slightly firm. Strain, and toss with a splash of oil.
Meanwhile, arrange the shrimp in a single layer on a large pie pan or paper plate, and pat them with a paper towel until completely dry. Season with salt and pepper.
Heat a large skillet over medium heat, and add 2 tablespoons of the butter. When the butter melts, raise the heat to medium-high, and invert the plate of shrimp over the skillet so the shrimp fall into the pan all at once. Cook the shrimp, without moving them, until the underside is pink, 1 to 2 minutes. Flip the shrimp, and cook until fully pink and cooked through, about 2 minutes more. Transfer the shrimp to a bowl.
Reduce the heat to medium, and add the remaining 6 tablespoons butter. Scrape the bottom of the skillet with a wooden spoon to release any browned bits. When the butter has mostly melted, whisk in the cream and nutmeg and bring to a simmer, then cook for 2 minutes. Lower the heat to keep the sauce warm.
Whisk the Parmigiano-Reggiano into the sauce. Add the shrimp and cooked pasta, and toss well. Season with salt and pepper. Serve hot in heated bowls.