The best burgers need soft and buttery brioche hamburger buns to create that perfect bite. The sourdough starter lends a subtle tang as well.
Mix all ingredients for the dough together except for the butter in a stand mixer. Mix on 1st speed for about 5 minutes. Turn the mixer on 2nd speed and mix until the dough comes together and pulls away from the side of the bowl. Turn the mixer back to 1st speed and add butter to incorporate. Turn back to 2nd speed and mix until the dough pulls away from the side of the bowl and is shiny and elastic.
Form the ball into a round shape and cover with plastic wrap. Let ferment all day or overnight at room temperature (about 8-10 hours), so it doubles in size. A sourdough culture ferments dough much more slowly than dry yeast, so you won't have to worry about it over-fermenting.
Take the dough out onto your work surface and pat into a rectangle. Cut into 8 equal-sized pieces. Flour your hands, but keep the work surface clean. Fold each piece of the dough toward the bottom to meet in the center. With the palm of your hand cupped, take the dough seam-side-down and begin to roll the dough into a ball so that it seals on the bottom. Use your hand as a guide, but don't press down on it so it forms a ball. When it's a firm ball, place on a parchment-lined sheet pan. Place up to six buns on one regularly-sized sheet pan.
Egg wash the tops of the buns and cover the pans with plastic wrap.
Proof the buns on top of the oven with the oven turned on so that it creates a warm space for the buns. Proof for about 45 minutes or until the buns have spread out a bit.
Egg wash the tops of the buns a second time before baking.
Bake in a 350 degree oven for about 30 minutes until the tops are golden brown. If you have two sheet pans, you can place one sheet pan in the fridge to slow down fermentation while the other pan is baking. Just make sure the buns remained covered in the fridge.
The best way to ensure the sourdough starter is ripe is to feed it 8-12 hours before making the dough. The starter will have risen and the top should be bubbly.