In the bowl of a stand mixer fitted with the dough hook, combine the flour, yeast, sugar, salt, and water. Crack in 5 of the eggs and beat on low speed for 3 to 4 minutes, until all the ingredients are combined, scraping down the sides of the bowl as needed. Knead the dough for 3 minutes more to make sure those ingredients are getting extra cozy.
Cut the butter into 1/2-inch cubes. With the mixer running on low, add the butter one piece at a time, until it works itself into the dough. This will take a few minutes. Perfect time to grab a coffee, check your e-mail, or stage a sweet Instagram snap. The dough will be quite wet at this stage, but fear not!
When all the butter has worked itself into the dough, turn the mixer to high and let it go to town, kneading the dough for 10 minutes. The side of the bowl should be clean, and the dough ball should be making a slapping sound against it.
Transfer the dough to a lightly oiled large bowl, cover, and let rise slowly in the fridge overnight.
The next day, let the dough sit out for 30 minutes or so to come to room temperature. Line two baking sheets with parchment paper.
Lightly dust a work surface with flour and place the dough on it. Using a kitchen scale, divide the dough into sixteen equal 3.15-ounce pieces (if need be, make 14 or 15 balls instead to get them to the right size). Using the palm of your hand, roll a piece of dough on the floured surface in a clockwise motion for about 1 minute. A perfect little dough ball will naturally form as it rolls on the counter. Pinch off the bottom of the dough ball to close it if necessary. As you finish each ball, place it on one of the prepared baking sheets, leaving at least 1 1/2 inches between each ball.
Cover the baking sheets with plastic wrap and let the dough rise until doubled in size, about 1 hour.
Preheat the oven to 350°F.
Crack and beat the remaining egg in a small dish. Brush the top of each dough ball with the egg, covering all sides. Repeat with a second coat of egg. You can either top each bun with 1 teaspoon sesame seeds or leave them plain. Choose your own adventure.
Bake for 30 to 35 minutes, until deeply golden brown. Let cool on a wire rack for a few minutes before serving. These brioche buns are best eaten the same day, but will keep in an airtight container at room temperature for 2 to 3 days.
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