In large nonstick skillet melt 2 teaspoons butter. Add onion. Cook, uncovered, over medium-low heat about 10 minutes or until onion is tender and beginning to brown, stirring occasionally.
Add mushrooms to onion. Cook, stirring occasionally, over medium-low heat for 3 to 4 minutes or until mushrooms are tender. Remove from skillet. Keep warm.
Add burgers to same skillet. Cook over medium heat, uncovered, for 7 to 8 minutes or until heated through, turning once. Remove from skillet. Keep warm.
Remove skillet from heat. Carefully wipe out skillet with paper towel. Lightly spread 2 teaspoons softened butter on one side of bread slices. Place two bread slices, butter side down, in skillet. Top with burgers, steak sauce, onion mixture and cheese. Place remaining bread slices, butter side up, on top. Cook, covered, over medium-low heat for 2 to 4 minutes or until golden brown on bottoms. Turn. Cook, uncovered, for 2 to 3 minutes more or until cheese melts and bread is golden brown. Cut in half. Serve immediately.