Activate Yeast: Add the warm water and milk to a small bowl of Pyrex (if milk is coming straight out of the fridge, pop it in the microwave for about 20 seconds). Add sugar and whisk to dissolve. Sprinkle yeast on top and activate. In about 10 minutes it should be bubbly and frothy.
Prep Flours: Add bread flour, all purpose flour, and salt to the bowl of a stand mixer. Give a quick whisk. Add softened butter and mix with the paddle attachment until all of the butter is well incorporated.
Create Dough: Crack the egg in a small bowl and whisk well with a fork. Remove the bowl of the stand mixer from the mixer (or tilt the head back). Once the yeast is activated add it with the flour and milk to the flours and add the whisked egg. Use a wooden spoon or another large spoon and give everything a few good mixes, ensuring your mixing up the flours from the bottom of bowl. Return bowl to the mixer.
Mix: Switch paddle attachment for the dough hook. Turn on mixer to level 4 (medium) and mix for 7 minutes. A very very sticky dough will form.
Proof: Grease a large mixing bowl with oil (an oil sprayer is helpful for this). Use a spatula to turn the sticky dough from the bowl of the stand mixer into the greased bowl. Cover loosely with either a lightly damp towel (see recipe notes), or loosely with plastic wrap. You don't want to fully seal the dough during this proof. Proof until doubled, 1-2 hrs.
Split Dough: Once the dough has doubled, turn it out onto a clean and lightly floured work surface. The dough will be very soft and slightly sticky, but not nearly as sticky as before it was proofed. Use a dough cutter and a kitchen scale to separate 8 evenly sized dough balls. Approximately 3.5 oz/100 g in size. Or split the dough in half, each half in half, and then those halves in half again until you have 8 doughs.
Shape into Balls: Add parchment paper or a silicone mat to the baking tray and set aside. To shape the dough into balls, first lightly flour your hands. Then flatten the dough in the palm of your hand into a rectangle shape. Pull out an edge of the dough away from your palm about 2-3 inches, then fold that back into the center. Repeat on the other corners of the dough, about 5 times. All of the folded-up corners will be layered in the center of the dough. Pinch the center together, rotating the dough as you pinch to remove the seams of those folded-in edges. This will create the surface tension of the dough ball that is needed for that signature smooth brioche bun top. Place the dough pinched side down, on the parchment paper. Repeat for all dough balls. Line parchment paper 3, 2, and 3 on the tray. (See photo in post)
Second Rise: Sprinkle a tiny bit of flour on the top of each of the dough balls and then cover gently with a (dry) tea towel or plastic wrap. Let rise a 2nd time until dough balls have almost doubled, 30-60 minutes.
Preheat: Preheat oven to 350℉ and place oven racks in the middle and lowest positions. Add an empty oven safe skillet or a rimmed baing tray to the lowest rack and allow to preheat with the oven.
Egg Wash: Add an egg to a small bowl and whisk well with a tablespoon of water. Use a pastry brush to gently brush the tops of the buns with the egg wash.
Bake: Add the tray of the buns to the middle rack of the oven, and then add 1 cup of water to the preheated skillet or rimmed baking tray underneath. Close the oven door quickly, as it will create a blast of steam, which you want for your buns! Bake for 20 minutes.
Cool and Enjoy! Remove from oven and place on cooling racks to let cool completely (or at least 30 minutes to allow the texture to set) before slicing and enjoying!