Place the flour, beaten egg, and breadcrumbs in three separate containers. Season the fish fillets with salt and pepper. Dredge the fish first in the flour, then the egg, then the breadcrumbs. Make sure to coat the fish evenly each time and to shake off extra flour or crumb. Place the breaded fish on a plate with a piece of waxed or parchment paper.
Heat a large skillet over medium flame for 2-3 minutes. Add the olive oil and allow the oil to heat for 20-30 seconds. Place the breaded fish carefully into the hot pan. Turn the flame down a bit and allow the fish to cook to a golden brown on one side and turn, about 2 minutes a side. Turn off the flame and remove the fish to a paper towel.
Mix the capers, anchovies, chopped parsley, lemon juice, lemon zest, and mayonnaise together in a small bowl. Serve the fish on a plate with a spoon of the salsa and a parsley sprig.