Get the butter and eggs out of the refrigerator and allow them to come to room temperature.
Using your quick-read thermometer, adjust your faucet temperature until it’s about 110° F, and then measure out a cup. Stir in the milk and sugar to feed the yeast. Sprinkle the yeast on top of the water and let it acclimate until it’s all damp.
Break an egg into a cup, beat with a dinner fork or whisk until smooth, and have it ready.
Measure or weigh the bread flour and all-purpose flour in the bowl of your stand mixer. Add the salt and softened butter. Run your mixer on low speed and let it stir the flour mixture for a minute or two until all the butter has disappeared unto crumbs.
Use a fork or a whisk to blend the yeast mixture until it is smooth. Keep the mixer running while you slowly trickle in the yeast mixture and the beaten egg.
Now speed the mixer up to a medium-low speed. I used speed 3. Run it until a dough forms, and then allow it to knead the dough for 5 to 8 minutes until it is smooth and flexible.If a lot of the dough continues to stick to the side of the bowl, stop the mixer and add a tablespoon of all-purpose flour. You’ll probably need to scrape the bowl and clear the paddle with a spatula a couple of times. Brioche dough is slightly softer and sticker than typical bread dough. A little bit will continue to stick to the bottom of the bowl. Just free it with your spatula.
Lightly spray the inside of a large bowl with cooking oil spray.
Put the dough on a floured counter and shape it into a ball with floured hands. Repeatedly stretch the top and tuck it under, turning the ball as you go, until the top is smooth and slightly stretched. Put the ball into the oiled bowl, hit it with a quick light spray of cooking oil, and cover the bowl with plastic wrap. Keep it someplace warm, preferably about 70° F. Wait 1 to 3 hours (or longer) until the ball has doubled in volume.
Cut a piece of parchment paper to fit your baking sheet.
Put a piece of wax paper on your scale and zero it. Deflate the ball of dough with floured hands and weigh it. Break the dough into eight pieces. Weigh and adjust each piece so they each weigh 1/8 the weight of the entire ball. You should have eight pieces of dough, all almost exactly the same size.
Roll each piece of dough into a ball in your hand, stretching the top, tucking it under, and turning the ball until the top is smooth. Place the balls of dough on the parchment-lined baking sheet. Spray a piece of wax paper lightly with cooking oil spray, and place it oiled side down over the dough balls to protect them while they rise. Keep them someplace warm for 1 to 2 hours (or longer) until the rolls have roughly doubled in volume.
Preheat your oven to 400° F. Put one shelf in the center and another shelf one level below it. Place a skillet or metal pan in the center of the lower shelf.
Break an egg into a cup and add about a tablespoon of cold water. Beat the egg mixture with a fork until smooth. Use a pastry brush to lightly coat each roll with the egg mixture.
Pour 1/2 cup of hot water into the pan on the lower shelf. Place the sheet of rolls in the oven just back of center. Bake 8 minutes.
After 8 minutes, turn the pan in the oven so it will get done evenly. If it looks like the rolls are going to burn, tent a piece of foil over them. Bake another 8 to 12 minutes until the rolls are golden brown.
When they’re cool enough to touch, move them to a wire rack to cool.