Top 2 of the bread slices each with a slice of the cheese. Divide and top the cheese with the meats. Top the meats with the last 2 slices of cheese and remaining two slices of bread. Heat the oil in non-stick skillet over medium heat. Dip the sandwiches on both sides in the beaten egg mixture. Cook 3 to 4 minutes per side, or until golden and hot in the middle. Cut each sandwich into quarters and serve.