Sauté onion in 1 tablespoon hot vegetable oil over medium heat 8 minutes or until tender.
Stir together onion, hash browns, salt, and pepper in a large bowl.
Heat 1 tablespoon oil in a large non-stick skillet over medium-high heat. Drop hash brown mixture into 3 1/2-inch rounds. Cook, in batches, 5 minutes on each side, or until lightly browned, adding remaining oil as necessary. Press down with a spatula to flatten; top each round with a cheese slice. Cook 5 minutes.