Add eggs, one at a time, beating well after each addition. Gradually add flour and mix well. Stir in the potatoes, Swiss cheese, onion, salt, pepper and cayenne pepper.
Heat butter and 2 tablespoons oil in a large nonstick skillet over medium heat. Drop batter by 1/4 cupfuls into butter and oil; press lightly to flatten. Fry in batches until golden brown on both sides, using remaining oil as needed.