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 the 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 over and switching and rotating position of baking sheets halfway through baking, until golden brown on both sides, about 15 minutes. Transfer onion rings to plates lined with paper towels to drain briefly. Serve immediately.