Adjust oven racks to lower-middle and upper-middle positions and heat oven to 450 degrees F. Place ¼ cup of the flour in a shallow baking dish. Whisk the egg and buttermilk together in a bowl. Whisk the remaining ¼ cup flour, salt, pepper and cayenne into the buttermilk mixture. Pulse the saltines and potato chips together in a food processor until finely ground; about 8 to 10 pulses. Place the crumb mixture in a second shallow baking dish.
Working one at a time, dredge each onion ring in flour, shaking off excess. Dip rings in buttermilk mixture, allowing excess to drip back into bowl, then drop into crumb coating, turning rings to coat evenly. Transfer coated onion rings to a large plate or tray.
Pour 3 tablespoons oil onto each of two rimmed baking sheets. Place the baking sheets in oven and heat until just smoking, about 8 minutes. Carefully tilt the heated sheets to coat evenly with oil, then arrange onion rings on the 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. Briefly drain onion rings on paper towel-lined plate. Serve immediately.
Note: The onion rings can be breaded in advance and refrigerated for up to an hour. Let them sit at room temperature for 30 minutes before baking them; if baked while still chilled from the refrigerator, the onions will not cook enough to soften and will remain crunchy.Nutritional values are based on one serving