Cook the fusilli according to package directions, omitting the salt if desired.
Meanwhile, sprinkle the chicken with 1/4 teaspoon of the salt and 1/8 teaspoon of the pepper. Spray a ridged grill pan with nonstick spray and set over medium-high heat. Add chicken and cook until cooked through, 5-6 minutes on each side. Transfer to a cutting board and let cool 5 minutes. Cut into 1/2 inch cubes and set aside.
Combine the half and half, milk, flour, nutmeg, and the remaining 1/2 teaspoon of salt and 1/8 teaspoon of pepper in a small bowl; beat with a whisk until blended.
Melt the butter in a large nonstick skillet over medium-high heat. Add the garlic and cook, stirring frequently, just until golden, about 1 minute. Whisk in the half and half mixture and cook, whisking constantly, until the mixture bubbles and thickens, 2-3 minutes. Stir in the fusilli, chicken, grated cheese, and parsley; cook, stirring gently, until heated through, about 1 minute longer. Serve at once with the shaved cheese (if using).