You will need 3 bowls. In one bowl mix together the bread crumbs, the grated parmesan cheese, and ½ of the shredded parmesan cheese, and mix well. In the second bowl four, in the third milk.
Pat dry the fish and season well with salt and pepper. Dip it in the flour, then the milk, and lastly, in the cheese mixture shaking off any excess mixture.
Sauté the fillet in a large skillet with 3 tablespoons of butter. Turn when nicely browned.
When both sides are done remove them from the stove and place them on a baking sheet. Divide the remainder of the cheese on top of each fillet and place it under the broiler for 2-3 minutes or until the cheese starts to bubble. When done remove from pan, place on plates, and keep warm.
Clean the pan from all butter leaving the brown bits. Add the remaining butter to the pan and over medium heat, melt the butter, add the lemon juice, and gently swirl the pan around to make a velvety sauce.
Garnish the fish with sauce and sprinkle some of the parsley if wish.