Chicken Fettuccine Alfredo

Kosher salt and freshly ground black pepper
12 ounces fettuccine
Olive oil, for tossing
12 ounces boneless, skinless chicken breast (about 2)
1 stick (8 tablespoons) unsalted butter
2 cups heavy cream
2 pinches freshly grated nutmeg
1 1/2 cups freshly grated Parmigiano-Reggiano