Kosher salt and freshly ground black pepperTo taste
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.
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. Cook, without moving, until the browned, 1 to 2 minutes. Turn the chicken, and cook until browned and cooked through, 2 to 3 minutes more. Transfer the chicken to a cutting board to rest for 10 minutes, then slice thinly.
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 lemon juice, and taste for seasoning. Add the chicken and cooked pasta and toss well. Season with salt and pepper. Serve hot.