Slice the onions into 5 mm (¼ inch) thick rings and separate out. Keep the larger rings and ditch the rest, or keep to fry up the next day for breakfast.
Make the batter next. Whisk together the egg yolk, oil, beer and flour in a bowl. Season with salt and pepper. In another clean, dry bowl, whisk the egg white until stiff, then gently fold into the batter until smooth and combined.
Heat 5 cm (2 inches) of vegetable oil in a deep heavy-based saucepan or a deep-fat fryer to 180°–190°C (350°–375°F), or until a cube of bread dropped into the oil browns in 30 seconds.
Dip the onion rings, a few at a time, into the batter, then carefully drop into the oil and deep-fry in batches for 1–2 minutes until golden. Remove very carefully using a slotted spoon and drain on kitchen paper. Serve at once with mayonnaise, while the onions are crisp and piping hot.