Gently heat the milk in a small saucepan or microwave safe container to 110°F. (30-45 seconds in the microwave.)
1 cup whole milk
Whisk together the warm milk, yeast, and sugar in the bowl of a stand mixer. Let sit for 5-10 minutes until foamy.
2 1/4 teaspoons active dry yeast, 2 tablespoons granulated sugar
Add the melted butter and egg, whisking briefly to combine.
3 tablespoons unsalted butter, 1 large eggs
Add the flour and salt. Using the dough hook attachment, mix on low speed until a shaggy dough forms. Increase to medium-low speed and knead for 6-8 minutes until the dough is smooth, soft, and slightly tacky. If needed, add a small amount of flour, 1 tablespoon at a time.
3 1/4 cups all-purpose flour, 1 1/4 teaspoons kosher salt
Lightly grease a large bowl with extra virgin olive oil. Transfer the dough to the bowl, cover, and let rise in a warm place for approximately 1-1 1/2 hours, or until doubled in size.
Extra virgin olive oil
Line a baking sheet with parchment paper.
Gently punch down the dough and transfer it to a lightly floured surface. Divide into 8 equal portions and shape each into a smooth ball. Arrange on the prepared baking sheet, spacing them several inches apart. Flatten each ball slightly with your palm.
Cover loosely and let rise for 45-60 minutes until puffy.
Preheat the oven to 375 degrees F.
Whisk together the remaining egg and water. Brush the tops of the buns with the egg wash and sprinkle with sesame seeds, if using.
1 large eggs, 1 tablespoon water, Sesame seeds
Bake for 15-18 minutes, rotating the pan halfway through, until deeply golden brown and cooked through.
Transfer the buns to a wire rack and cool completely before slicing.
Notes
If mixing by hand: Use a dough whisk or sturdy spoon until you have a shaggy dough, then knead by hand on a flour-dusted countertop for 6-8 minutes, until smooth.