Cook rigatoni in water with the tablespoon salt until 1 minute short of the package time. Reserve 1 cup pasta water before draining.
Meanwhile, season chicken with its salt and pepper. Heat oil in a large skillet and cook chicken for 7 to 8 minutes until browned and at least 165°F; transfer to a plate.
Cook garlic in the skillet for 30 seconds, add tomatoes and cook for 5 to 6 minutes until they split, pressing a few gently with a spoon. Add beans, broth and rosemary and simmer for 3 minutes.
Return chicken and add rigatoni, spinach and 1/4 cup reserved pasta water. Toss over low heat for 1 to 2 minutes until the spinach wilts and the pasta is tender.
Stir in Parmesan off the heat and loosen with more pasta water if needed so the tomato juices coat the tubes. Serve promptly.