In 12-inch skillet, heat oil and 1 tablespoon butter over medium-high heat. Add onions; cook about 5 minutes, stirring frequently, until onions begin to brown. Reduce heat to medium-low and cook 20 to 25 minutes, stirring frequently, until onions are golden brown.
Add broth, wine, salt and pepper. Cook, stirring occasionally, for 8 to 10 minutes, until the liquid has evaporated. Add vinegar and cook, stirring, for 1 to 2 minutes, until onions are dark brown. Transfer onions to bowl; cool 10 minutes.
Spread 2 teaspoons butter on one side of each slice of bread. On the unbuttered side of 4 bread slices, place 1/3 cup Gruyère and 1/3 cup Cheddar cheese. Top each with 1/3 cup onion mixture and 1 slice remaining bread, buttered side out. Place Parmesan cheese in shallow dish; carefully press both the buttered sides of each sandwich into Parmesan cheese to make cheese stick frimly to bread.
Heat 12-inch nonstick skillet over medium heat. Add 2 sandwiches; cook uncovered 4 to 5 minutes or until bottoms are golden brown. Carefully turn; cook 2 to 3 minutes longer or until bottoms are golden brown and cheeses are melted. Repeat with remaining 2 sandwiches.
Nutrition information per serving: