Put the flour in a wide shallow bowl. Season the cutlets all over with 1 tsp. salt and 1 tsp. pepper. Dredge in the flour and shake off any excess.
Heat the oil in a 12-inch skillet over medium-high heat until shimmering hot. Working in batches if necessary, cook the cutlets until dark golden-brown, 1 to 2 minutes per side. Transfer to a large plate as they finish.
In the same skillet over medium heat, melt the butter. Add the onion and garlic and cook, stirring often, until softened, about 3 minutes.
Add the mushrooms and cook, stirring occasionally, until they release their liquid and it reduces to a glaze, about 5 minutes. Stir in the tomatoes, wine, and rosemary. Bring back to a simmer, scraping up any browned bits from the bottom of the skillet. Reduce the heat to medium low and simmer until the sauce is thickened, about 7 minutes. Season to taste with salt and pepper.
Add the cutlets and any accumulated juices to the sauce and cook until heated through, 1 to 2 minutes. Serve immediately.