1. Prepare the Dough:
In the bowl of a stand mixer fitted with the dough hook, combine the flour, sugar, salt, and yeast. Add the eggs and milk. Mix on low speed until a shaggy dough forms.

2. Add the Butter:
Gradually add the softened butter, one cube at a time, allowing each piece to fully incorporate before adding the next. Knead on medium speed for 8–10 minutes, or until the dough is smooth, elastic, and slightly sticky.

3. First Rise:
Transfer the dough to a greased bowl, cover with plastic wrap, and let it rise in a warm spot for 1–2 hours, or until doubled in size.

4. Shape the Brioche:
Punch down the dough and turn it onto a floured surface. Shape into a loaf or divide into smaller portions for rolls. Place in a greased loaf pan or on a lined baking sheet.

5. Second Rise:
Cover with a clean towel and let rise again for 45 minutes to 1 hour, or until puffy and slightly jiggly to the touch.

6. Prepare the Egg Wash:
In a small bowl, whisk together the egg and milk. Brush the mixture over the surface of the brioche.

7. Bake:
Preheat your oven to 350°F (175°C). Bake for 25–30 minutes for a loaf, or 15–20 minutes for rolls, until golden brown and the bread sounds hollow when tapped on the bottom.

8. Cool and Enjoy:
Allow the brioche to cool on a wire rack before slicing or serving.