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.