01 - Combine flour, sugar, yeast, and salt in the bowl of a stand mixer fitted with a dough hook.
02 - Add room temperature eggs and lukewarm milk to the dry ingredients. Mix on low speed until a shaggy dough forms.
03 - Increase mixer speed to medium and knead for 5 minutes until the dough becomes smooth and begins to develop structure.
04 - Gradually add softened butter cubes, a few at a time, mixing thoroughly after each addition. Continue kneading for 10 minutes until the dough is glossy, elastic, and pulls away from the bowl sides.
05 - Shape dough into a ball, place in a lightly greased bowl, cover, and let rise in a warm area for 1 to 1.5 hours until doubled in size.
06 - Gently deflate dough and transfer to a lightly floured surface. Divide into three equal portions, roll each into a rope, and braid together. Place in a buttered 9x5-inch loaf pan.
07 - Cover loosely with plastic wrap and let rise for 45 minutes until puffy and nearly doubled in volume.
08 - Preheat oven to 350°F.
09 - Whisk egg yolk with 1 tablespoon milk until combined. Gently brush the mixture over the top of the risen brioche.
10 - Bake for 25 to 30 minutes until golden brown and a skewer inserted into the center comes out clean.
11 - Let cool in the pan for 10 minutes, then transfer to a wire rack to cool completely before slicing.