Carefully slice open the vanilla pod using a sharp knife and scrape out the seeds. You can save the vanilla pod to make vanilla sugar.
In a large bowl whisk the eggs, full-fat milk, vanilla seeds and nutmeg. Add the brioche slices and allow to soak for a few minutes. Turn over and continue to soak.
Heat a little of the oil and butter in a non-stick frying pan over a medium heat until it starts to foam.
Carefully drain two slices of the eggy bread with a slotted spatula and transfer to the pan. Fry the bread for approximately three minutes on each side, until golden brown.
Remove and keep warm placed on kitchen paper, on a baking tray in a warm oven. Repeat the process and cook the remaining bread.