Warm the milk, water, and butter. In a glass measuring cup, put the water, milk, and unsalted butter. Microwave on low for 45 seconds to soften the butter and slightly warm the milk and water.
Mix all ingredients. Add the contents of the measuring cup to your bread machine bucket or stand mixer bowl. Then add the remaining dough ingredients in the order listed. Start with the lesser amount of flour, adding more as needed. The dough will be sticky but should form a ball.
Knead. Set machine for DOUGH cycle, or knead in the stand mixer 8-10 minutes on low. It is difficult to knead brioche by hand, so use a machine if you have one.
The first rise. Allow the dough to rise for two hours, tightly covered, until very puffy.
Divide and shape. Divide dough into 8-12 pieces, using electric scale if you are particular. You will get 8 very large buns or 12 medium buns. Line two baking sheets with parchment paper. Shape dough pieces into balls and place them on a baking sheet. Brush rolls with egg wash and sprinkle with seeds if desired. Cover tightly with greased plastic wrap.
The second rise. Allow the buns to rise for one hour at warm room temperature. Towards the end of the rising time, preheat the oven to 400 degrees.
Bake. Bake for 16-18 minutes, until very golden brown. Cool on wire rack. If you'd like, toast before using as burger buns