Rinse chicken and pat dry; remove and discard excess fat.
Pour olive oil into a 5- to 6-quart nonstick pan over high heat; when hot, add chicken and cook, turning once, until browned on both sides, 5 to 7 minutes total. Transfer to a plate.
Add onion, carrots, mushrooms, and garlic to pan; stir often until mushrooms begin to brown, 8 to 10 minutes. Add tomatoes and their juices, tomato sauce, rosemary, and the chicken with any accumulated juices; bring to a simmer, cover, reduce heat to low, and cook for 5 minutes. Turn chicken over and simmer, covered, until no longer pink in center of thickest part (cut to test), about 5 minutes longer. Season to taste with salt and pepper.
Meanwhile, in a 3- to 4-quart pan, stir broth and polenta until well blended. Bring to a boil over high heat, stirring often; reduce heat to low and stir often until polenta is creamy and smooth to bite, 8 to 12 minutes (about 3 minutes for cornmeal).
Spoon polenta into wide bowls or onto rimmed plates. Top with equal portions of chicken and sauce; sprinkle with parsley. Add parmesan cheese and more salt and pepper to taste.