Add 1 cup flour, yeast, salt and sugar to a stand mixer bowl; whisk to combine and set aside.
In a saucepan or microwave-safe dish, heat water, milk and oil to 120-130°F.
Add warm liquids to flour mixture; mix with paddle attachment for 4 minutes on medium speed. Add egg; beat 1 minute.
Switch to dough hook attachment. Gradually add enough remaining flour to make a soft dough, and knead until smooth and elastic, about 5 to 7 minutes.(NOTE: For hand mixing: following above steps, mix ingredients in large mixing bowl using a wooden spoon or dough whisk. Knead on floured surface until smooth and elastic.)
Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until indentation remains after poking dough with finger down to second knuckle, about 60-90 minutes.
Turn dough onto lightly floured surface; punch down to remove air bubbles. Divide into two equal pieces. Divide each half into 3 or 4 pieces. (See Note for hotdog buns option.) Shape each piece into a smooth ball. Place on large greased baking pan. Flatten each ball to a 4-inch diameter, allowing one inch between buns. Cover; let rise until indentation remains after touching, about 20 minutes.
Brush egg wash over buns. Sprinkle tops with sesame seeds (optional).
Preheat oven to 400°F.
Bake for 10 to 12 minutes. Remove from baking pan and let cool on wire rack.