Bring a large pot of salted water to a boil. Cook the capellini according to the package directions, usually about 2–4 minutes, until just al dente. Reserve ½ cup pasta water before draining.
Season the small chicken pieces with Italian seasoning, garlic powder, black pepper, and ½ teaspoon salt.
Heat the olive oil in a skillet over medium-high heat. Add the chicken and cook for 5–7 minutes until golden-brown with lightly caramelized edges and 165°F inside. Add 1 tablespoon butter and half the minced garlic, toss briefly, then transfer to a plate.
Melt the remaining butter in the skillet. Add the remaining garlic and stir for 30 seconds. Pour in the heavy cream and milk, simmering gently for 2–3 minutes until slightly thickened.
Remove from the heat and gradually stir in ¾ cup Parmesan and the mozzarella until smooth. Add the capellini and gently lift and turn the delicate strands through the sauce, loosening with a little reserved pasta water if necessary.
Serve the capellini loosely spread across one side of a white plate, spooning the remaining sauce over the noodles. Arrange the small garlic Parmesan chicken pieces beside it and finish with the remaining Parmesan and parsley.