Combine flour, kosher salt, and freshly ground black pepper in a small bowl.
Place each piece of meat between two sheets of plastic wrap and gently pound meat to 1/4-inchthickness with flat side of a meat pounder, turning meat over occasionally.
Lightly coat each piece of meat with seasoned flour.
Heat olive oil in a wide 5- to 6-quart heavy pot over moderately high heat until hot but not smoking. Sauté meat (working in batches if necessary), stirring constantly and turning over, until evenly and lightly browned on both sides, 5 to 6 minutes.
Remove meat from pan and set aside.
Add onion, bell pepper, celery, and mushrooms and cook, stirring and scraping up any brown bits from bottom and side of pot, until softened, 5 to 6 minutes.
Add tomatoes and garlic and cook, stirring frequently and scraping bottom and side of pot, 3 minutes.
Return meat to pan.
Add bay leaves, thyme, rosemary, salt, cayenne, black pepper, broth, and wine, then reduce heat and simmer, partially covered, stirring occasionally, until meat is very tender, about 1 1/2 hours. Taste broth, and add balsamic vinegar to cut the richness of the braising liquid if desired.
