Make 4 crosswise incisions in the onions, taking care not to cut all the way through the root. Place the onions in the ice-cold water, leave to soak for approx. 45 mins. Drain the onions and pat dry.
Combine the mayonnaise with all the other ingredients up to and including the salt. Cover and chill the dip.
In a deep dish, mix the flour with all the other ingredients up to and including the salt.
Combine the milk and eggs in a bowl. Toss the onions in the spiced flour, dip in the batter, toss once more in the flour.
Fill a pot to ⅓ with oil and heat it to approx. 170°C. Using a slotted spoon, lower the onions into the oil in batches so that the roots face upwards. Deep-fry the onions for approx. 3 mins., turn and fry for a further 3 mins. Remove and drain on paper towels. Serve the onions with the dip.