Scoop a ball of ice cream and wrap completely with the slices of cake, patching where needed with smaller pieces.
Wrap the balls tightly with plastic wrap, forming a round shape. Freeze at least 5 hours or overnight.
Heat the oil in a deep fryer or heavy pan.
Stir cold sparkling water very slowly into the flour; whisk in the egg to create the batter.
Remove the ice cream balls, one at the time, from the freezer. Dust the surface with flour. Dip into the batter, coating completely. Quickly lower coated ice cream balls into the oil; fry until golden, about 25 seconds. Remove with a slotted spoon.
Serve with a drizzle of chocolate sauce before ice cream melts!