Whisk together the flour, cornmeal, beer, egg, and salt in a medium bowl, making sure to break up any clumps. Cover the bowl, then place in a cooler for at least 20 minutes and up to 3 hours.
Pour the oil into a large Dutch oven. Bring to a temperature of 375 degrees F (190 degrees C).
Dry the fish fillets with paper towels, then season both sides with salt and pepper. Place each piece of fish into the bowl of batter and make sure fully they are covered.
When the oil is heated, take one piece of fish out of the batter, letting the excess batter drip back into the bowl. Then, gently lower the fish into the hot oil. Repeat the process for all fillets. Cook until the fillets are golden brown, about 5 minutes.
Let rest on a paper towel for 1 minute for the oil to drain, then serve hot.