For the chicken: Preheat the oven to 350 degrees F.
Sprinkle the chicken breast with the Italian seasoning, salt and pepper. Heat the oil in a pan and cook the chicken on medium heat until it registers 165 degrees F in the middle, about 5 minutes on each side. Cube the chicken and set aside.
For the alfredo sauce: In the same pan that the chicken was cooked in, melt the butter over low heat. Add the minced garlic and cook until the garlic has softened, about 2 minutes. Add the flour and whisk until smooth. (Be careful not to brown the roux.) Add the milk and heavy cream a little at a time, making sure to whisk until smooth after each addition, and let thicken, about 5 minutes. Add the nutmeg and pepper. Whisk to combine. Turn off the heat and mix in 2 1/2 cups Parmesan. Stir until all the cheese is melted. Add salt and pepper to taste.
Layer 1 cup of the alfredo sauce on the bottom of a 9-by-13-inch baking pan. Add 3 lasagna noodles. Spread another cup of alfredo, a third of the chicken, a sprinkle of parsley and a quarter of the mozzarella. Then add 3 more noodles and repeat the layering two more times, ending with alfredo sauce, the remaining mozzarella and 1/2 cup shredded Parmesan.
Bake until browned on top, 40 to 50 minutes. Remove the lasagna from the oven and sprinkle with parsley. Let the lasagna rest for 10 minutes before cutting into it.
You can skip the chicken portion of the recipe and swap in the breast meat from 1 rotisserie chicken, cubed (you need about 3 cups of meat).