Heat the oven to 400°F. Lightly grease 8 cups of a standard muffin pan.
In a large bowl, combine the ground chicken, egg, breadcrumbs, Parmesan, milk, parsley, garlic powder, Italian seasoning, salt, and black pepper. Mix gently until just combined.
Divide the chicken mixture into 8 portions. Press half of each portion into the bottom of a prepared muffin cup. Place a mozzarella cube in the center, then cover it with the remaining chicken mixture. Pinch the edges together so the cheese is enclosed.
Bake for 20 to 25 minutes, until the muffins are cooked through and the centers reach 165°F. Let them rest in the pan for 5 minutes before removing.
While the muffins rest, heat the olive oil in a large skillet over medium heat. Add the zucchini noodles and sauté for 2 to 3 minutes, tossing gently, until just tender. Divide the noodles among four plates and serve with two chicken muffins per plate.