Warm the Ingredients: Combine 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.
Mixing time: 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 and a tablespoon, 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 for a low setting of 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.
The first rise. Place the dough in a greased bowl and cover it securely with plastic wrap. Allow it to rise in a warm, draft-free location 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.
The 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.
Baking. Before baking the buns, make sure that your oven is preheated 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.