In a large bowl, mix the warm milk, yeast, sugar, and two cups of flour using a stand mixer or a sturdy spoon. Stir well, then cover the bowl with a towel and leave it in a warm place for around 30 minutes.

In a separate bowl, whisk the eggs until smooth. Add the whisked eggs, salt, and melted butter to the dough mixture. Gradually incorporate the remaining flour into the dough, a little at a time. Depending on the flour, you may need an extra cup.

Knead the dough by hand or with the dough hook of a stand mixer until it’s smooth and elastic, which should take about 10 to 15 minutes.

Cover the dough with a towel and let it rise in a warm spot for about 1 hour.

After the dough has risen, divide it into 18 to 24 portions, depending on your preferred bun size. Shape each portion into a round, flat bun.

Place the buns onto a baking sheet lined with parchment paper. Cover them with a towel and let them rise again for 30 minutes. Meanwhile, whisk together an egg with 3 tablespoons of water in a small bowl, then brush this mixture onto the tops of the buns.

Bake the buns in a preheated oven at 350°F for about 20 minutes, or until they turn a rich golden brown. Enjoy your freshly baked brioche buns!