Preheat oven to 450 degrees F.
Whisk together egg whites, buttermilk, flour, salt, and cayenne in a shallow bowl until smooth.
Separate onion slices into rings and pick out the 12 largest, reserving remainder for another use.
Heat 1-inch of oil in a 12-inch nonstick skillet over moderately high heat until hot but not smoking. Working in batches of 4, dredge onions in flour and dip rings into batter, letting excess drip off, then cook in skillet until golden brown, about 1 minute on each side. Transfer rings as browned to a lightly oiled, shallow (1-inch deep) baking pan.
Bake rings in middle of oven, turning once, until crisp and deep golden, 12 to 15 minutes total.
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