In the bowl of a stand mixer fitted with the dough hook, combine 3 1/2 cups of the flour with the yeast, sugar, salt, softened butter, eggs, and milk. If mixing by hand, stir everything together with a wooden spoon in a large bowl, then turn the dough out onto a lightly floured surface.

Knead the dough on low speed for 10 minutes, or knead by hand for the same amount of time.

Gradually add the remaining flour, 1 tablespoon at a time, only as needed until the dough no longer clings to the sides of the mixer bowl. For hand kneading, add flour a little at a time until the dough feels tacky but not sticky and is easy to handle without leaving residue on your hands.

Transfer the dough to a lightly oiled bowl, cover with plastic wrap or a damp tea towel, and let it rise in a draft-free spot for 60 to 90 minutes, or until it nearly doubles in size.

Turn the dough out onto a clean work surface and divide it into 12 equal portions. For perfectly even buns, weigh each piece to about 3 ounces (85 g).

Shape each portion into a smooth ball, then flatten it gently with your thumbs and forefingers into a round about 3 inches wide.

Arrange the rounds on a lightly floured surface, cover them with a clean tea towel, and let them rest for 15 minutes so the dough relaxes and becomes more elastic.

After the rest, stretch and flatten each round again until it measures 4 to 4 1/2 inches in diameter. Place the rounds at least 3 inches apart on parchment-lined baking sheets. If you are using dark-colored baking sheets, reduce the oven temperature by 25 degrees F and line the sheets with heavy-duty aluminum foil or two layers of standard foil to prevent over-browned bottoms.

Cover the baking sheets with clean tea towels and let the buns rise for another 60 to 90 minutes, until they are puffy and roughly doubled in size.

Preheat the oven to 350 degrees F.

Whisk the reserved egg with 1 tablespoon of water to make the egg wash. Brush the mixture lightly and evenly over the tops of the risen buns.

Sprinkle sesame seeds evenly over the buns, if using.

If desired, score the tops gently with a razor blade or bread lame. This step is optional but can encourage a better rise in the oven.

Bake the buns for 15 to 20 minutes, until the tops are evenly golden brown. After 15 minutes, use an egg turner to lift one bun and check the bottom. If the bottom is already deeply browned, remove the pan from the oven even if the tops are still pale; the buns are likely fully baked when the bottoms are well browned.

Let the buns cool completely on a wire rack before slicing or serving.