Cut pork into 1/2 in. slices; flatten to 1/8 in. thickness. Coat with Parmesan cheese; set aside.
In a large skillet, saute the onion, mushrooms and garlic in 1 teaspoon butter and 1 teaspoon oil until tender; remove and keep warm. In the same skillet, cook pork in remaining 1 teaspoon butter and 1 teaspoon oil in batches over medium heat until juices run clear, about 2 minutes on each side. Remove and keep warm.
Add broth to pan, scraping to loosen browned bits. Stir in wine or additional broth; add seasonings. Bring to a boil. Reduce heat; simmer, uncovered, for 5 minutes. Combine cornstarch and water until smooth; stir into pan juices. Bring to a boil; cook and stir until thickened, about 2 minutes. Serve with the pork and the onion mixture.