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