Warm the milk with the butter in a small saucepan. Set to the side.
In a large bowl, mix flour, yeast, sugar, vanilla sugar and 2 eggs. Separate the third egg and add the white and half of the yolk to the mix. Reserve the remaining yolk in a cup. Knead with the dough hook of an electric mixer to a smooth, elastic dough. Cover and let rise in a warm place until doubled in size, about 30 minutes.
Take about 1/5 of the dough and shape the large and the small portion into a ball. Place the large ball on a lined baking tray. Gently push the bottom of a cup or a glass into the centre of the ball to make a 2cm deep dent.
Stir the reserved egg yolk with 1 teaspoon milk and brush the dough ball with it all over. Place the small ball in the dent; brush the entire loaf with egg wash. Let rise while preheating the oven.
Preheat the oven to 200 C / Gas 6. Bake the brioche in the preheated oven until golden brown, about 20 to 25 minutes.