In the bowl of a stand mixer fitted with the dough hook attachment, stir together the warm water, sugar, and yeast until combined. Let it stand for about 5 minutes, or until foamy.
To the same bowl with the yeast mixture, add the flour, butter, salt, and 1 of the eggs. Knead the mixture on medium-low speed for 10 minutes, or until the dough is smooth and elastic.
Scrape the dough down into a ball, and lightly oil the top of the dough and sides of the bowl. Cover the bowl with plastic wrap and let it rise in a warm, draft-free spot (75°F) until it doubles in size, 1 to 1½ hours.
Punch down dough and turn it on to a lightly-floured surface. Divide the dough into 8 equal pieces. Pinch the corners of each piece together at the center, then flip them over. On a clean surface, cup the ball with your hands and move it in a tight circular motion to shape each bun. (This will create a tight ball with a smooth top.)
On a large, parchment-lined, rimmed baking sheet, place the buns seam-side down, about 3 inches apart.
Using the palm of your hand, flatten each ball to be 3 inches wide. Loosely cover the baking sheet with plastic wrap and let the rolls rise in a warm, draft-free spot (75°F) until visibly puffed, 45 minutes to 1 hour.
After 25 minutes of proofing, preheat the oven to 375°F.
Once the rolls have finished rising, in a small bowl, whisk together 1 tablespoon of water and the remaining 1 egg until combined. Brush the tops of the buns with the egg wash, then sprinkle them evenly with the sesame seeds.
Bake for about 15 minutes or until golden brown. Remove the buns from the baking sheet and let them cool completely on a wire rack, for about 30 minutes.
Notes
Weigh the flour. Too much flour will ruin homemade burger buns. The most effective way to measure flour is to weigh it with a kitchen scale. If you don’t have one, fluff the flour up in its container, spoon it into a measuring cup, and level off the top with the straight side of a knife.
Punch down the dough. This is a vital step when making any kind of yeast bread. Punching down the dough helps redistribute the yeast and also helps remove excess air pockets for a more fine crumb. To punch down dough, make a fist and gently push down into the center of the dough after its first rise to deflate it.
Dough rises best in warm, draft-free spots. Let the dough rise in an area that reaches about 75°F for the best results. You can use your oven on the “bread proof” setting or inside with the light on.
If using your oven’s proofing setting: Pull the rolls out about 15 to 20 minutes before the proof time is up, begin preheating the oven, and set the baking sheet with the dough balls on the stovetop.
For a flavorful twist: Use everything bagel seasoning or dried minced onion and poppy seeds in place of sesame seeds. Or, mix black and white sesame seeds together for a colorful variation.