Using a small sharp pointed knife or a pastry cutter make a whole in the brioche slices.
In a large bowl whisk two of the eggs, the full-fat milk, sea salt and freshly ground black pepper. 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 them to the pan. Fry the bread for approximately three minutes, until golden brown and flip over.
Crack in an egg into the hole in each slice. Gently cook for another three minutes until the bread is crisp and the egg is cooked. Spoon hot butter, from the pan, over the egg to aid cooking.
Remove and keep warm on a baking tray in a warm oven. Repeat the process and cook the remaining bread.
Sprinkle with the Parmesan if using and serve.
