Bring a large pot of water to a boil, and salt generously. Add the pasta and cook according to package directions until al dente (tender but still slightly firm). Drain and toss with a splash of oil.
Meanwhile, slice the chicken into 1/4-inch-thick strips, and lay them on a plate or a sheet of waxed paper. Season with salt and pepper.
Heat a large skillet over medium heat. Add 2 tablespoons of the butter. When the butter melts, raise the heat to medium-high and add the chicken in 1 layer. Cook, without moving the pieces, until the underside has browned, 1 to 2 minutes. Flip the pieces, and cook until browned and cooked through, 2 to 3 minutes more. Transfer the chicken to a medium bowl.
Reduce the heat to medium. 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 just warm.
Whisk the Parmigiano-Reggiano into the sauce. Add the chicken and cooked pasta and toss well. Season with salt and pepper. Serve hot in heated bowls.
Don't worry if it seems like there is too much sauce. As soon as everything is tossed together, the sauce will start to cling to the pasta and thicken before your eyes!
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