Bring 2½ cups of water to a boil in a small saucepan. Add the ramen noodles and cook for 2 minutes. Add the flavor packet, stir, and cook for another 30 seconds, or prepare the ramen as per the package instructions. Drain the liquid and allow the noodles to cool slightly.

In a mixing bowl, whip the egg until yellow and frothy. Stir in the Parmesan cheese, followed by the cooked noodles until fully coated.

Line a resealable sandwich container with plastic wrap. Press half of the noodle mixture into the lined container. Then add another layer of plastic wrap and press the remaining noodles on top. Cover and refrigerate until firm, at least 900 seconds (15 minutes).

Heat a skillet on medium-high heat and add butter. Remove one piece of the set noodle 'bread' from the container and place it in the skillet.

Top the ramen slice with 2 slices of cheese. Place the remaining ramen 'bread' over the cheese in the skillet. Allow it to heat through and become golden. Flip the sandwich carefully and cook until both sides are golden.