It's impossible to beat a burger served on a soft, pillowy brioche bun. This recipe (from Hewn bakery in Evanston, Illinois) makes a big batch, so you can keep a stash in the freezer for several meals.
Place milk, 4 eggs, the flour and yeast in the bowl of a stand mixer fitted with the dough hook attachment. Mix on medium speed until dough is smooth and comes together, about 3 minutes, scraping down the bowl after the first minute of mixing, if needed. Let dough rest in bowl 20 minutes to help relax the gluten.
Add sugar and salt; mix on low speed 3 minutes more, scraping dough off the hook after the first minute.
Add half of the butter and continue to mix on low until butter cubes are no longer visible and are completely incorporated into dough. Add remaining butter and mix until incorporated.
Once all the butter is added, increase speed to medium and mix until dough makes a slapping sound and comes away from the bowl, about 16 minutes. Be patient and resist the urge to increase the speed. You will start hearing a distinct slapping sound when dough is almost ready.
Transfer dough to a medium bowl coated with cooking spray. Cover with plastic wrap; let stand at room temperature 1 hour, then punch down dough to help redistribute the yeast. Cover again and refrigerate at least 2 hours or overnight.
After dough has completed its 2-hour rest in the refrigerator, grease 2 large baking sheets or line them with parchment paper.
Divide dough into fifteen pieces. (If you have a kitchen scale, they will weigh about 68 grams apiece.) Take each piece and place in the palm of your hand. Round them on the counter to create skin tension, lightly flouring your hand as needed. Place balls on baking sheets, 2 to 3 inches apart. Lightly cover with plastic wrap.
Preheat oven to lowest setting, about 170°. Place a cake pan on lowest oven rack. Add 1 cup boiling water to the pan. Turn off oven. Place pans with buns on center oven rack. Let proof until risen by about 1/2 inch, 21/2 to 4 hours. Remove buns from oven and remove plastic wrap. Remove cake pan with water from oven. Preheat oven to 350°.
In a small bowl, whisk together eggs and the water. Gently brush each bun with egg wash. Bake until golden brown, rotating pans halfway through baking, about 20 minutes. Let cool at least 10 minutes, or cool completely and cover with plastic wrap to use the next day. The buns can be frozen for up to 3 months in a resealable plastic bag. Warm frozen rolls in the oven at 350° for 10 minutes before using.