First, wrap the brie in the prosciutto slices and set it aside. Turn the chilled brioche dough out onto a lightly floured surface and roll it into a 25cm circle. Place the prosciutto-wrapped brie in the center of the dough, then carefully fold the edges of the dough over the brie, sealing it neatly into a parcel.
Transfer the dough parcel to a baking tray lined with baking parchment. Brush the top and sides generously with beaten egg, then chill it in the fridge for 30 minutes. After chilling, brush it with another layer of beaten egg and chill for a further 30 minutes. Finally, let the parcel rise at room temperature for 1 hour.
Preheat your oven to 200°C/180°C fan/gas 6. Bake the brioche for 22 minutes, until the crust is golden brown and the cheese is molten inside. Serve it warm, straight from the oven.