Place butter in a small saucepan over medium heat. Cook until it melts. Be sure to keep a close eye on it as butter burns easily. Turn the heat off and let it cool to room temperature.
Meanwhile, whisk together bread flour, instant yeast, and salt in a large bowl. Set aside.
In a separate bowl, whisk together 6 large eggs, water, and sugar.
Pour the egg mixture into dry ingredients along with the now-cooled butter. Using a wooden spoon, stir to combine. It will look like it is too wet and almost like there is more liquid than there should be. That is normal. Do not overmix. Mix it until no dry flour remains and a uniform mass forms. This should take about 1 minute.
Cover the bowl with plastic wrap and let it rest for 10 minutes.
Holding the edge of the dough with your fingertips, fold it over onto itself halfway towards the middle. Turn the bowl 45 degrees and repeat the same process. Do this 6 more times. It should be 8 times in total. Cover with plastic wrap and let it sit for 30 minutes.
Repeat folding and rising every 30 minutes three more times. After the fourth set of folds, cover it tightly with plastic wrap and place it in the fridge for 16-48 hours.
Line two sheet pans with parchment paper. Set them aside.
Transfer the dough onto a lightly floured surface. Divide the dough into 12 equal pieces*. If you have a kitchen scale, you can use it to weight each piece to ensure rolls equal size rolls. Or, you can use a ruler as a guide to cut them.
Working with one piece at a time, pat dough into a 3-inch disk. Holding the edge of the dough, fold the edges of the dough onto itself halfway towards the center. Repeat folding until it forms a ball. Flip it over and move your hands in small circular motions to form it into a smooth round ball. This dough might be sticky so feel free to sprinkle it with a bit of flour if it sticks to your hands. Repeat the same process with the rest of the balls.
Place them in the prepared sheet pans. I recommend 6 brioche buns per sheet pan with them being 3-inches apart from each other. Be aware that they do get larger as they rest and after they are baked.
Cover them with plastic wrap and let them rise for 2 hours or until they “almost” double in size. If you are baking these brioche buns during the summer months, they rise faster. And if you are making it during the colder months, it might be a little longer. I recommend placing them in a warmer part of your kitchen to speed up the process.
Thirty minutes prior to baking, preheat your oven to 350 degrees F.
Make an egg wash by whisking together one egg with a tablespoon of water in a small bowl. Remove the plastic and brush each brioche roll generously with the egg wash. If using, sprinkle them with sesame seeds.
Place both sheet pans into the oven. Bake for 20-25 minutes until they turn golden brown on top (or until the internal temperature reaches 190 degrees F.) making sure to rotate the baking sheets halfway through the baking time.
Take them out of the oven and let them rest for 5 minutes, let them cool on a wire rack, or serve while they are still warm.