Preheat oven to 180 degrees. Bring a pot of salted water to a boil. Add in the pasta, and cook as stated on the package. Drain pasta and set aside.
On medium heat in a medium-size pot start the sauce by adding in the butter, (let it melt) Then slowly add in the flour, stir consistently. Continue by slowly adding in the milk, nutmeg, egg, salt, pepper, and the three tablespoons of parmesan cheese. Bring heat up to high and whisk to get rid of any lumps.
Take a rectangular baking pan and layer the bottom part with one thick layer of bechamel sauce. Then add a layer of pasta and top with cheeses, continue the process until the top is layered with bechamel sauce.
In a small bowl combine the breadcrumbs with ½ cup of parmesan cheese. Gradually garnish the bechamel sauce until it is fully covered. Bake at 180 for 40 minutes. Once it is baked set it aside to cool before cutting into the Pastitsio