Max Rocha's Deep-Fried Bread & Butter Pudding with Cold Custard

For the pudding:

6 plain brioche buns, halved

4 eggs

370ml milk

180g caster sugar, plus extra for dusting

370ml cream

sunflower oil, for deep-frying

1 quantity custard (see recipe), chilled, to serve

For the custard:

330ml milk

330ml cream

1 vanilla bean, split lengthways and seeds scraped

160g egg yolks (9–10 eggs)

170g caster sugar

Makes 750ml