Drain tomatoes, reserving oil mixture. Cut tomatoes into thin strips; set aside. In a large skillet, saute mushrooms and onion in butter and olive oil for 5 minutes or until mushrooms are browned. Stir in the vinegar, reserved tomatoes and oil mixture. Bring to a boil. Reduce heat to low; cook for 10 minutes.
Arrange salad greens on a platter; top with mushroom mixture and cheese.