1/2 cup all-purpose flour
1 large egg, at room temperature
1/2 cup buttermilk, at room temperature
1/4 teaspoon cayenne pepper
Table salt and ground black pepper
30 Saltines
4 cups kettle-cooked potato chips
2 large yellow onions, cut into 1/2-inch-thick rounds
6 tablespoons vegetable oil
Adjust oven racks to lower-middle and upper-middle positions and heat oven to 450 degrees. Place 1/4 cup flour in shallow baking dish. Beat egg and buttermilk together in medium bowl. Whisk remaining 1/4 cup flour, cayenne, 1/2 teaspoon salt, and 1/4 teaspoon pepper into buttermilk mixture. Pulse saltines and chips together in food processor until finely ground and place in separate shallow baking dish.
Pull apart rings in each round, discarding any that are less than 2 inches in diameter. Working one at a time, dredge each onion ring in flour, shaking off excess. Dip in buttermilk mixture, allowing excess to drip back into bowl, then drop into crumb coating, turning ring to coat evenly. Transfer to large plate and repeat with remaining onion rings.
Pour 3 tablespoons oil onto each of two rimmed baking sheets. Place in oven and heat until just smoking, about 8 minutes. Carefully tilt heated sheets to coat evenly with oil, then arrange onion rings on sheets. Bake, flipping onion rings and switching and rotating position of baking sheets halfway through baking, until golden brown on both sides, about 15 minutes. Transfer onion rings to paper towel-lined plates to drain briefly. Serve immediately.