"My mom fell in love with these medallions of beef when she lived in California. When the restaurant was about to close, the owner took her back to the kitchen and taught her how to make them. We beg for them on all special occasions! Serve over wide egg noodles."
Melt butter in a large skillet over medium heat. Cook beef slices in butter until browned completely, about 5 minutes per side. Remove beef slices to a plate, reserving butter and drippings in the skillet.
Saute mushrooms and onion in the butter mixture until onions are clear, 5 to 7 minutes. Push the vegetables to the edge of the skillet.
Stir red wine and beef baste into the butter mixture in the skillet; cook and stir, adding more beef base as needed, until the sauce is thickened, 7 to 10 minutes.
Arrange beef slices into the gravy. Top each beef slice with a slice of ham. Top the ham with Monterey Jack cheese. Put a lid on the skillet and cook until the cheese melts, about 5 minutes.
Remove medallions to a platter. Spoon sauce and vegetables over the medallions.