Place oil in a deep skillet until it comes about 1-inch up the side of the pan. Heat oil to 360 degrees F. Mix buttermilk, egg, and hot sauce in a medium bowl. Mix flour, cayenne, 1 teaspoon salt, and pepper in a bag. Pat onion slices dry and dip in the buttermilk mixture to coat. Transfer slices to bag and toss to coat. Deep-fry onions in batches until golden, about 2 minutes. Transfer to a paper towel-lined plate to drain, sprinkle immediately with salt, and serve.