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Blend the butter with the parsley, Parmesan cheese, garlic and pepper until well combined. Spread the butter mixture evenly over one side of each slice of bread. Place a slice of Cheddar, on the unbuttered side, of the other 2 slices of bread. Toss the mozzarella with the Gruyere and sprinkle over the Cheddar. Top the sandwiches with the remaining bread, buttered-side-out.
Turn the grill on. Select the program Panini and press OK. Once the purple indicator light has stopped flashing, place the sandwiches on the grill and close the lid.
Cook until the indicator light has changed to red or until the cheese is melted and the bread is toasted.