Preheat oven to 500°F.

Combine bread crumbs, cornmeal, salt, and cayenne in a large sealable plastic bag and shake to mix.

Season fish with salt and pepper on both sides and, working with 1 piece at a time, put fish in bag and shake to coat well with crumbs. Dip fish in eggs, then shake in crumbs again to coat. Transfer fish to a plate.

Heat 3 tablespoons oil in a 12-inch heavy ovenproof skillet (preferably cast-iron) over high heat until hot but not smoking, then fry fish until undersides are golden brown, about 1 minute. Turn over, add remaining 3 tablespoons oil, and cook 1 minute more. Put skillet in upper third of oven and bake until fish is just cooked through, about 5 minutes.