Brown butter (This can be made in advance)

1. In a small stainless steel or light-interior colored pot, add 8 of the 9 tablespoons of butter over medium-low heat and melt until beautiful golden brown bits are visible and the butter smells nutty. Do not walk away from this because you can easily burn/blacken the butter. The bubbles will appear large in the beginning as the water evaporates, and it will be loud. Then it will settle to a finer, quieter, thicker foam. This is when the butter starts to brown very quickly. Swirl it around using the handle of the pot or use a spoon if you can’t see what’s happening. You’re looking for caramel colored bits, and once you see that, immediately remove the butter from the heat and add the remaining 1 tablespoon of the 9-total tablespoons to the browned butter to help it cool down and prevent it from over-browning. **Caution: Please be aware that when the remaining tablespoon is added, it will foam up quite a bit so please be careful that it doesn’t spill over or cause any burns to your skin. 
2. Pour the browned butter into a small heat-proof bowl or measuring cup and cool to room temperature for 30 minutes. Once the butter is cooled down, cover the bowl or cup with plastic wrap and pop it into the fridge for about an hour or until it has firmed up enough to handle.
3. Scoop the cold browned butter onto the center of a decent-sized piece of plastic wrap. Fold one piece of the plastic over the browned butter to form it into a log shape (it does NOT have to be perfect, it’s not that serious). Using the plastic wrap, press and gently roll it just until its in about a 5 to 6’inch sized log. Snuggly wrap and secure the butter plastic, ensuring there are no air-pockets and pop it back into the freezer for another half hour. At this point, you can leave the browned butter in the freezer for a few days, but if you do, make sure you keep it in an airtight container or freezer bag. 
4. About 20-30 minutes before it’s time to add the browned butter to the dough, remove from freezer, cut it into 1-inch chunks and let it sit on the counter just to take the chill off. 

To make the dough

1. Lightly grease the inside of a large bowl and set aside. Line two half-sheet baking pans with a silicone mat or parchment paper, and set aside. In a small bowl, beat 1 large egg with 1 tablespoon of water and set aside. Portion out the sesame seeds, if using, and set aside. 
2. In the bowl of your stand mixer fitted with the dough hook, add the flour and yeast and give it a quick mix with a whisk until combined. 
3. In a large cup or small bowl, beat 2 large eggs into the water to combine. With the mixer on low speed, slowly add the water/egg mixture. Add the 2 beaten eggs and water to the flour mixture. Mix the flour, yeast, eggs and water mixture on low speed (speed number 2 on your mixer) just until it starts to come together. This will take about 2 minutes. Scrape down the sides of the bowl frequently to make sure everything is getting incorporated. Once it’s all combined, turn off the mixer, remove the dough hook and cover the bowl with plastic wrap. Let the dough rest for 15 minutes. This gives it time to hydrate and bloom the yeast. After 15 minutes, remove the plastic wrap and reattach the dough hook. 
4. Add the granulated sugar and salt to the dough and knead on medium-low speed until the salt and sugar is fully incorporated, about 45 seconds. The salt is added at this stage so that it doesn’t inhibit the growth of the yeast. So make sure you don’t accidentally add the salt before this step. 
5. Increase the speed to medium and alternately add the browned butter chunks and remaining 4 tablespoons of cubed/room temp butter every 30 seconds until all of the butter is absorbed into the dough, about 3 minutes. When there are no obvious knobs of unmixed of butter, continue to knead the dough for about 10 minutes longer. The dough will be tacky, but it shouldn’t be sticky in an un-workable way. Please watch the above “how to” video for visual reference. Troubleshoot: There shouldn’t be any wet/loose dough pooling at the bottom of the bowl, but if there is, incrementally add more bread flour 1-tablespoon at a time, waiting about 2 minutes between each addition until it comes together.
6. The dough is ready when it’s soft and supple to the touch. It may have a bit of tackiness but it shouldn’t be sticking to your hands too much at all. Using a bowl scraper or rubber spatula, transfer the dough to the lightly greased bowl and cover with plastic wrap to rest for 1 hour or until doubled in size. My favorite place to let dough rest is in a turned-off oven with just the oven light on.
7. After the dough has doubled in size, turn it out onto your counter. You shouldn’t need to add any flour at this point because there’s enough fat in the dough to keep it from sticking. 
8. Using kitchen shears or a metal dough scraper, divide the dough into 12 pieces. Working one piece at a time, gather each ball of dough into a ball, using your fingers to pinch the seams closed. Then cup your hands over the ball and apply gentle pressure to quickly cup-and-roll each ball into a smooth shape. Your hands will make very brisk tightly-closed circular motions to form the dough. You are not pressing down! The pressure is light. Repeat until every ball is formed and the seams are closed.
9. Gently transfer 6 balls of dough to each lined baking sheet, 3 per side lengthwise. Very lightly grease 2 large pieces of plastic wrap and LOOSELY drape it over each pan of buns. Rest the dough on your counter for 1 hour or until doubled in size. About 30 minutes into allowing the dough to rest, preheat your oven to 350 degrees F. 
10. Make the egg wash: beat 1 large egg with a tablespoon of water and a pinch of salt until well combined. After the dough has rested for an hour, carefully remove the plastic wrap. If you see any crazy air bubbles on the surface, gently pop with a toothpick. Brush each bun with the egg wash and quickly sprinkle with a desired amount of sesame seeds, if using. 
11. Bake one sheet at a time on the center rack of the oven for 18-20 minutes, rotating half-way through. The buns should be a deep golden brown and register between 205-210 degrees F on an instant-read thermometer.
12. Cool on a wire rack for a half hour before serving, or store the completely cooled buns in an airtight container or zipper-lock bag for a couple of days. They can be frozen for up to a month. Enjoy.