Separate one egg and put the white into one bowl and the yolk and the whole egg into a second bowl. Add the flour and half the caster sugar to the second bowl. Whisk the egg white until if forms soft peaks, then use the same whisk to beat the flour mixture until smooth, gradually whisking in the milk. Fold in the egg white.
Pour the oil into a deep, heavy-based saucepan until it comes one-third of the way up the side, then heat until it reaches 180–190°C (350–375°F), or until a cube of bread browns in 30 seconds. Dip a few apple slices in the batter and turn gently to coat. Lift out one slice at a time and lower carefully into the oil. Deep-fry, in batches, for 2–3 minutes, turning until evenly golden. Remove with a slotted spoon and drain on kitchen paper.
Meanwhile, put the blackberries, remaining sugar and measurement water in a small saucepan and heat for 2–3 minutes until hot. Arrange the fritters on serving plates, spoon the blackberry sauce around and dust with a little icing sugar.