In a small saucepan, whisk together the TangZhong ingredients: water, milk, and flour. Whisk well to smooth out any lumps. Set the pan over low heat, and whisk constantly until the mixture thickens -- about 4 minutes. Remove from heat.
Whisk in each addition to the TangZhong: milk, egg, melted butter, and yeast.
In a large (-ish) bowl, stir in the listed dry ingredients for the dough. Then stir in the TangZhong mixture. Stir until a shaggy dough develops. On a lightly-floured board, knead the dough by hand until it is elastic and fairly smooth -- about 5 minutes. Wash out, dry, and lightly grease the mixing bowl.
Form the dough into a ball, and place it in the greased mixing bowl. Then flip the dough to grease its underside. Cover with cling film, and then place in a warm location. Let rise until doubled (or nearly doubled) in volume -- 90 minutes to 2 hours.
Lightly grease a 9-inch-diameter cake pan. Gently deflate the dough. Then cut the dough in half. Cut each half into 4 equal size pieces to achieve 8 pieces total. Form each dough segment into a roll, and place in the prepared cake pan. Cover and let rise until the rolls fill the pan -- about 50 minutes. Meanwhile, preheat the oven to 350°F.
Brush the rolls with egg wash (1 egg beaten with 1 teaspoon water). Brush gently to avoid deflating the rolls. Top the rolls, if you wish, with a sprinkling of sesame seeds.
Bake in the preheated oven until the rolls are brown and beautiful -- 25 to 30 minutes. Let cool for at least 10 minutes in the pan before serving.