Melt butter and 1 tablespoon oil in a nonstick skillet over medium heat. Combine potatoes, salt, and pepper.
Heat 1 tablespoon vegetable oil in a nonstick skillet over medium-high heat. Drop potato mixture into 3 1/2 inch rounds; cook, in batches, 5 minutes on each side or until golden, adding remaining oil as necessary. Press down with a spatula to flatten; top hash browns evenly with Jarlsberg or Monterey Jack cheese during the last 5 minutes.