In a small bowl, combine mayonnaise and horseradish; cover and refrigerate until serving. In another small bowl, combine Parmesan cheese, 1 tablespoon butter and garlic powder; spread over bun tops. Set aside.
In a large bowl, combine the beef, steak sauce and onion soup mix. Shape into four patties.
Moisten a paper towel with cooking oil; using long-handled tongs, lightly coat the grill rack. Grill burgers, covered, over medium heat or broil 4 in. from the heat for 4-5 minutes on each side or until a thermometer reads 160° and juices run clear.
Top with Swiss cheese; cover and grill 1-2 minutes longer or until cheese is melted. Place buns, cut side down, on grill for 1-2 minutes or until toasted.
Meanwhile, in a large skillet, saute mushrooms and green onions in remaining butter until tender. Serve burgers on buns; top with horseradish sauce, French-fried onions, mushroom mixture and, if desired, tomato and lettuce.