Heat the peanut oil in a 5-quart Dutch oven over high heat until it reaches 350ºF on a deep-fry thermometer. Adjust the heat to maintain temperature.
Whisk the cornmeal and flour together in a shallow dish. Combine the Old Bay, kosher salt, paprika, and pepper in a small bowl. Season the catfish fillets evenly on both sides with the spice mixture. Pour the buttermilk into another shallow dish. Dip each fillet into the buttermilk, flip once to coat both sides, hold over the pan, and allow the excess to drip off. Coat both sides of the fillets in the cornmeal mixture. Set the coated fillets on a wire rack and let rest for 5 minutes.
Gently add the fillets, two at a time, to the hot oil, and fry until golden brown, 5 to 6 minutes. Using a spider strainer, carefully remove the fried fillets to a wire rack set over a paper towel-lined half sheet pan. Repeat method with remaining fillets. Arrange the catfish on a serving platter and serve immediately.