In the bowl of a stand mixer, combine warm water, warm milk, yeast, sugar, and salt. Let it sit for 5 minutes to activate the yeast.
Add in softened butter, one egg, and half the flour. Stir until combined.
Switch to the dough hook and knead, adding the remaining flour ½ cup at a time until the dough is soft, smooth, and tacky. Knead for about 10 minutes, until it passes the windowpane test.
Lightly coat the dough in oil, cover, and let it rise in a warm spot for 1 ½ to 2 hours until doubled in size.
Deflate the dough and transfer it to a lightly oiled surface. Divide into 8 equal pieces. Now shape each piece into a ball by gently pulling the edges into the center and pinching them together. Flip it over so the seam is on the bottom. Cup your hands around the dough ball and roll it in small circles on the counter to create a smooth, round shape.
Place buns on a parchment-lined baking sheet, spacing them 3 inches apart. Press each bun to slightly flatten. If they resist, wait 5 minutes, then press again to about 3 to 3 ½ inches wide.
Loosely cover and let rise until puffy, about 30 minutes, not doubled. Meanwhile preheat the oven to 375°F.
Mix the remaining egg with 1 tablespoon water together in a small bowl. Then gently brush each bun. Sprinkle with sesame seeds.
Bake at 375°F for 15 minutes or until golden brown and hollow sounding when tapped. Cool completely before slicing.