Preheat the oven to 375 degrees F (190 degrees C). Grease a casserole dish with cooking spray.

Bring a large pot of unsalted water to a boil. Add penne. Cook, stirring occasionally, until tender yet firm to the bite, about 11 minutes; add spinach to boiling water for the last minute. Drain and set aside.

Grease a large nonstick skillet with cooking spray and place over medium-high heat. Cook and stir chicken and basil in the hot skillet for 3 minutes. Stir in tomatoes and spaghetti sauce; bring to a boil. Simmer on low heat until chicken is cooked through, stirring occasionally, about 3 minutes. Stir in Neufchâtel cheese until melted.

Add penne-spinach mixture and 1/2 cup mozzarella cheese to the skillet; stir until well combined. Pour into the prepared casserole dish.

Bake in the preheated oven until heated through, about 20 minutes. Top with remaining 1/2 cup mozzarella cheese and Parmesan. Continue baking until mozzarella is melted, about 3 minutes.