Preheat the oven to 375°F (190°C).
Bring a large pot of water to a boil and cook the penne according to package directions; omit added salt. Add the spinach to the boiling water during the last minute of cooking, then drain the pasta and spinach in a colander and return to the pot.
While the pasta cooks, spray a large nonstick skillet with cooking spray and heat over medium-high. Add the bite-size chicken pieces and dried basil and cook, stirring, about 3 minutes until the chicken is mostly cooked through.
Add the spaghetti sauce, minced garlic, and drained diced tomatoes to the skillet; bring to a boil, then reduce heat and simmer about 3 minutes or until the chicken is fully cooked. Stir in the cubed Neufchâtel cheese until melted and evenly incorporated.
Stir the chicken-and-sauce mixture and 1/2 cup of the shredded mozzarella into the drained pasta and spinach until combined, then transfer the mixture to a 2-quart casserole or 8-inch square baking dish.
Bake in the preheated oven for 20 minutes. Remove from oven, sprinkle with the remaining mozzarella and the grated Parmesan, then return to oven and bake an additional 3 minutes or until the mozzarella is melted.