Warm the milk and add the yeast. Let it rest for 10 minutes.
In the mixer bowl, fitted with the dough hook, combine the flour, sugar, vanilla sugar, and salt.
Mix well, then create a well in the center.
Add the egg yolks and milk.
Mix and knead until the mixture is smooth.

Add the butter, one piece at a time, and knead until the dough is soft and elastic, about 8 to 10 minutes.

Cover the bowl with plastic wrap and let it rise for 2 hours.

On a floured surface, divide the dough into 3 equal parts. Roll each part into a rope and braid, making sure to tuck the ends under the braid.
Place the braid on a baking sheet lined with parchment paper, then cover with a clean cloth.
Let it rise for 2 hours.

Preheat the oven to 350°F (180°C).
Mix the egg yolk and 35% cream, then brush the brioche with the mixture.
Sprinkle with granulated sugar.
Bake for 35 to 40 minutes, or until the brioche is golden brown and the bottom sounds hollow when tapped.