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 try. Gently push the bottom of a cup or a glass into the centre of the ball to make a 2cm deep depression.
Stir the reserved egg yolk with 1 teaspoon milk and brush the dough ball with it all over. Place the small ball in the depression and also brush with the egg wash all over. 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.