Mix water, yeast and sugar together. Let sit for 5-10 minutes until mixture becomes bubbly.
Add egg, egg yolk, butter, salt, and one cup of flour. Mix together.
Continue adding flour 1 cup at a time until the dough forms a ball. Once you reach 4 ½ cups of flour, add any remaining flour in small amounts. The amount of flour used will vary.
Knead in a stand mixer at a low speed for 5 minutes, or for 10 minutes by hand.
Cover and let rise until doubled, 1 ½ - 2 hours.
Gently deflate and divide dough in half. Working with one half at a time, divide into equal size pieces.
Roll each piece into a round shape by using your hand cupped around the dough and making a circular motion.
Flatter each ball by gently pressing down on the top. Let the buns rest for a few minutes and then press again.
Place on a baking sheet covered with parchment paper, or lightly sprayed with a non-stick spray.
Cover and let rise again for 30-60 minutes until slightly puffy.
Make an egg wash by beating the reserved egg white with water. Use a pastry brush and brush buns with the egg wash. Sprinkle sesame seeds across the top.
Bake in a preheated 350°F oven for 18-22 minutes until golden brown.
Remove from baking sheet and cool on a baking rack. Allow the buns to fully cool before slicing.