Step 1Pat both sides of each filet dry with paper towels and season lightly with salt and pepper. If they are very large, cut in half widthwise on the diagonal. The ideal serving size is 3 to 4 ounces per filet.
Step 2Place about 1 cup of flour in the middle of a large dinner plate. Beat the egg with the water in a large flat-bottomed bowl. Put about 2 cups of bread crumbs in the middle of another large dinner plate.
Step 3Working 1 filet at a time, lay the fish in the flour and dust both sides, shaking off any excess. Then dip the fish in the egg wash, coating both sides and wiping with your fingers to remove any excess. Bury the filet in the bread crumbs and press lightly so they adhere. Set the filet aside on a wire baking rack placed over a baking sheet. Repeat to bread all of the filets. (The recipe can be prepared up to this point and refrigerated on the rack over the baking sheet for up to 6 hours.)
Step 4When ready to cook, melt 4 tablespoons of butter in a large skillet over medium-high heat. When the butter has melted and finished bubbling, dip a corner of 1 filet into it; it should immediately begin to sizzle. If it does not, remove the fish and let the butter heat a little more. This should only take 20 to 30 seconds more; do not let the butter burn.
Step 5When the butter is hot enough, lay each filet in the skillet, being careful not to crowd the fish. Cook until the underside is golden brown, 4 to 5 minutes. Turn the filets; cook until you can easily slip the point of a sharp knife into the center, 3 to 4 minutes more. Remove to a platter lined with paper towels and keep warm.
Step 6Wipe the pan clean with a paper towel. Melt the remaining 2 tablespoons of butter in the skillet over high heat. When it sizzles, whisk in the lemon juice and parsley.
Step 7Remove the fish from the oven, salt it lightly, and put it on a serving plate. Drizzle with the lemon butter. Serve immediately.