Bring a large pot of salted water to a boil and cook the penne according to package directions until al dente; drain and set aside.
In a large skillet over medium heat, combine the jarred pasta sauce, pressed garlic, and red pepper flakes; heat until the sauce is bubbling and fragrant.
Add the cooked chicken to the sauce and heat until the chicken is warmed through.
Stir in the cottage cheese until incorporated and the sauce is creamy.
Toss the drained pasta into the skillet with the sauce and chicken, then fold in the shredded mozzarella until melted and combined; season with salt and pepper to taste.
Serve hot and top with additional Parmesan if desired.