In a small bowl, combine water, milk, yeast, and sugar. Allow to sit about 10 minutes. The mixture will start to look foamy on top. This means the yeast is active!
Add in egg and butter.
In a stand mixer, combine flour (starting with 3 cups) and salt. Using a dough hook attachment, mix together the flour and salt mixture with the yeast mixture. Allow to knead on medium speed for 3 to 5 minutes. If dough seems too sticky, add 1/4 cup of flour at a time. You do not want a dry dough, but it shouldn't be sticking to the sides of the bowl much.
Shape into a ball and place into a greased bowl. Cover with a towel and let sit in a warm place until doubled in size. May take 1 to 2 hours.
Line a baking sheet with parchment paper.
Punch out air in the dough and divide into 8 equal pieces. Roll each one into a ball.
Place on parchment paper and cover with a towel. Allow to rise until doubled.
Place a baking dish, filled halfway with water, on the bottom rack of oven. Preheat to 400 degrees F.
Mix together the egg and water for the egg wash. Brush thoroughly over each bun.
Bake for 15 minutes, or until tops are golden brown. Enjoy!