Whisk bread flour, all purpose flour, yeast, sugar, and salt together until well combined.*
Using a hand or stand mixer, add in softened butter and beat on medium until the butter is thoroughly incorporated and the dough looks just slightly crumbly.
Add the beaten egg, warm milk, and warm water. Use a stiff wooden utensil handle and mix in all ingredients to combine and form a dough.*
Knead for 8-10 minutes by hand until it all comes together and the glutens work their magic. This process is a little sticky and will get all over your hands, but is easily washed off. If the dough is not sticky add 1 Tablespoon of warm water and knead it in with your fingers until it gets a little stickier. (It should not be a stiff, shaggy ball of dough. You want it sticky and pliable.)
Shape dough into a ball in the same bowl, cover the bowl with plastic wrap, and let sit until it doubles in size, around 1 hour. Do not rush this process. Make sure your dough is nice and puffy and has definitely doubled. (Tip: I like to turn on my oven to 250 for about 4 minutes, turn the oven off, then pop the bowl in there and let it sit until it is done rising. This gives it a nice warm environment to rise faster. Just remember to turn the oven OFF)
After it has risen, sprinkle a cutting board with flour and get that sucker out of the bowl. Flour a spatula and work it out. Knead the dough about 7-8 times.
Using a dough scraper or knife, cut the dough ball in half, then each in half, then each in half again to form 8 pieces. If you want smaller buns you can cut them in half one more time.*
Take each ball, flatten it like a pancake on the floured surface, then pull the sides up into the center rotating around to get a tight ball, pinching them in the middle to seal the seams. (I will get a picture of this process the next time I make these). Then flip them over seam side down on an unfloured surface, cup your hand over it, and roll in a small circle. Don’t be afraid of it; use a little pressure. This will round out the bun into a perfect shape. (If it sticks to your hands while you do this you won’t get a good smooth round shape on the top, so lightly flour your hands or the dough ball before rolling it)
Transfer to a parchment lined baking sheet, about 3 inches apart. Lightly cover with plastic wrap and let rise for 1 hour, or until slightly puffier (see photos above). Preheat oven to 400° and add an empty small metal pan (do not use glass) to the bottom rack of your oven.
10. After they have risen, (optional) mix 1 egg with 3 Tablespoons water. Brush the tops with the egg wash and (also optional) sprinkle with sesame seeds if using.
11. Pop your buns in the oven, add 1/2 cup cold water to your empty pans and shut the oven door right away. This will make steam and keep the buns nice and moist.
12. Bake for 15-18 minutes until golden brown on the top.
13. Take them out and cool on a rack. Enjoy!