Mix the yeast with the warm milk, and leave a few seconds.
Put in a bowl of a Kitchenaid miwer, and add the flour, the eggs, the sugar and the salt.
Mix for 7 minutes at minimal speed, then 7 minutes at speed 2.
Add softened butter and knead again for 7 minutes at 1st speed. Then, knead at speed 2 until a ball forms (it can take around 20 minutes).
Cover the dough and leave to swell in the fridge for at least 2 hours
Take the dough and knead it to degas.
Cut it in 8 pieces of same weight, and form balls. Grease a cake mould, and place the balls in it. Put the mould in a warm place and leave to swell for 2 hours.
Mix the yolk and the milk together, and brush the brioche with it.
Bake for 30 minutes at 360°F.
Take out the oven and leave the brioche to cool down.
There you are, your brioche is ready !