For the Tangzhong or water roux. Add all the ingredients to a pan. Over low heat, keep stirring until slightly thickened and turn into a smooth paste. Set aside and let it cool.
Proof the Yeast. In a small bowl add milk and mix in the active dry yeast and maple syrup. Let it sit for 5 to 10 minutes until foamy.
Knead the dough. In a bowl of a stand mixer, add all the dry ingredients. Mix with the dough hook attachment until blended. Add the proofed yeast mixture and eggs. Mix on low speed and add the softened butter in 4 additions then mix in the Tangzhong mixture. Knead for about 8 to 10 minutes or until the dough is smooth. Knead by hand for about 12 minutes or until the gluten developed. This dough should be sticky and elastic.
Test to check the dough when kneaded enough. Windowpane test by stretching without tearing, which means the gluten has been developed.
Proof the dough. With a spatula, scrape the bowl. Drizzle with vegetable oil, grease the bowl and the dough, cover with a kitchen towel. Place in a warm place and let it rise for an hour to 2 hours or until it doubles in size.
Divide the dough. Grease your hands with vegetable oil Punch out the gas to deflate the dough and divide it into two evenly by weighing on the kitchen scale. To be exact, each dough weighs about 700 grams each.
Divide and shape the dough for burger buns. Divide the dough evenly into 6 with the kitchen scale to be exact. Each dough weighs between 112 grams to 117 grams. Roll the dough by capping with your oiled hands and forming it into a ball. Place seams down on a baking sheet with parchment paper at least two inches apart. Cover with a kitchen towel and let it rest to rise for about 45 minutes to an hour, or until it doubles in size.
Divide and shape the dough for hotdog buns. Divide the dough evenly into 6 with the kitchen scale to be exact. Each dough weighs between 112 grams to 117 grams. For smaller buns divide into 8. Each dough weighs between 85 grams to 90 grams. Use a rolling pin to flatten the dough, then roll it out to 5 to 6 inches in width. Roll tightly into a cylindrical shape. Pinch the seams then roll to smooth out the dough. Place 2 inches apart on a baking sheet with parchment paper. 2nd proof. Cover with a kitchen towel and let it rest to rise for about 45 minutes to an hour, or until it doubles in size.
Brush the dough with egg wash. Mix the egg and water in a small bowl. Generously brush each dough with eggwash. Sprinkle the top of each dough with optional sesame seeds.
Bake the proofed dough. Slide in the middle rack and bake in a preheated oven at 350°F for about 15 to 18 minutes or until golden brown. Transfer on a cooling rack and let it cool.