To make the eggy bread, pour the custard into a large bowl.
Add the brioche to the custard and mix to coat.
Melt 25g/1oz butter in a medium frying pan over a medium heat. Add the custard-coated brioche and cook for two minutes on each side, until golden.
To make the caramelised apples, place 50g/1¾oz of butter and the sugar in a medium frying pan over a medium heat, until melted together.
Add the apples and sultanas and stir well to coat and caramelise.
Add the lemon juice and the rum. Warm through and carefully set light to the rum to flambé for a few seconds.
To serve, place the eggy bread on a warm plate and garnish with the caramelised apples and sultanas.