In a large mixing bowl, combine the flour, sugar, salt, and yeast.
Add the warm milk and eggs to the dry ingredients, and mix until a dough forms.
Knead the dough for about 10 minutes, or until it becomes smooth and elastic.
Gradually add the softened butter to the dough, and continue kneading until it is fully incorporated.
Cover the dough with a clean kitchen towel, and let it rise in a warm place for about 1-2 hours, or until it doubles in size.
Punch down the dough to remove any air bubbles, and shape it into a loaf.
Place the loaf on a baking sheet lined with parchment paper.
Brush the loaf with the beaten egg wash.
Preheat the oven to 375°F (190°C).
Bake the brioche for about 30-35 minutes, or until it is golden brown.
Remove the brioche from the oven, and let it cool before slicing and serving.
