Warm the milk. In a bowl, dissolve the yeast with 100 ml of lukewarm milk and leave it to rest for 15 minutes.
In a bowl, mix the flour with salt and sugar. Sink a well. Pour the dissolved yeast, the rest of the lukewarm milk, and 1 beaten egg into the well. Mix with your fingertips, bringing the flour to the center of the bowl, until all the ingredients are incorporated.
Place the dough on a floured work surface and knead for about 5 minutes. Add the butter cut into small pieces and knead again. The dough should be smooth, homogeneous, and slightly sticky. Form a ball, place it in the bowl and cover with a tea towel. Leave to rise for one hour and half hours in a warm place.
Knead the dough with your fist to remove the air. Cut the dough into three pieces and form sausages with each piece and braid them. Place the braid on a baking sheet covered with parchment paper. Cover with a tea towel and let rise for 1 hour in a warm place.
Preheat the oven to 180°C/356°F. Using a pastry brush, brush the top of the brioche with the second beaten egg. Bake for 25 minutes. Remove from the oven and let it cool on a wire rack before slicing.