Warm. Combine the water, milk, and butter in a glass measuring cup. Microwave on low for about 45 seconds to melt the butter without overheating the milk—a perfect lukewarm temperature.
Mix dough. Pour the lukewarm mixture into your bread machine's or stand mixer's bowl. Add the eggs, sugar, 4¼ cups of flour, salt, and yeast. Begin mixing on a low setting. The dough will initially be sticky; add more flour a tablespoon at a time, if necessary, until it forms a cohesive, slightly tacky ball.
Knead. Choose the DOUGH cycle on the machine or knead the dough in a stand mixer on a low setting for 8-10 minutes. Make sure that the dough is smooth, stretchy, and slightly sticky to the touch. Be careful not to add too much flour while kneading, as brioche dough is naturally soft.
First rise. Place the dough in a greased bowl and cover it securely with plastic wrap. Let it rise in a warm, draft-free place for 2 hours or until it has doubled in size and appears fluffy. This gradual rise enhances the taste and texture of the dough.
Divide and shape. Gently press the dough to release the air, and then divide it into either 8 or 12 equal parts, depending on the size of the buns you want. To shape the dough, pull it down and under to make a smooth ball. Put the shaped buns on baking sheets lined with parchment paper, and make sure to leave enough room between them so they can expand.
Second rise. Gently apply a beaten egg on top of the buns using a brush, and add sesame or poppy seeds if desired. After that, cover the trays with lightly greased plastic wrap and keep them in a warm place for about an hour or until they have nearly doubled in size.
Bake. Preheat your oven to 400°F (204°C). Bake the buns for around 16-18 minutes, rotating the baking sheet halfway through for even browning. You can tell that the buns are ready when they turn a deep golden color.