1.Warm the milk in the microwave for 30 seconds then add 1 tablespoon water. Stir, and let sit until it feels lukewarm. Stir in the yeast and 1/8 teaspoon sugar.
2.Let the yeast mixture bloom, about 5 minutes. It should be foamy on the surface. Add to the bowl of a stand mixer with dough hook. In a small bowl, whisk together the flour with the salt and 1 tablespoon sugar.
3.To the milk and yeast mixture, stir in the butter and egg. (Don't worry if the butter doesn't melt all the way immediately). Beat briefly to combine.
4.While the mixer is running, add the dry ingredients in two batches. Crank the mixer to medium-high, and knead for 8-10 minutes. The dough will be sticky.
5.Place the dough in a well-oiled bowl, turning it to coat with oil. Cover loosely with plastic wrap, and then let rise in a warm place for about 2 hours, or until doubled. (Rising time is dependent on the temperature of the room; let rise until the dough is doubled).
6.Punch the dough down very well, and gather it into a ball. Weigh the dough, and divide it by 6--each dough ball should be between 73 and 74 grams. Roll each dough piece into a ball, and then use your fingers to pinch the edges under until the seam is on the bottom. Then, roll the dough between your hands to make a log shape about 4" long.
7.Repeat with the other 5 pieces of dough. Place the 6 dough sticks in a 9x13" pan lined with parchment paper, and cover loosely with plastic wrap. Space the dough logs about an inch apart--they will rise and touch each other slightly. Cover with plastic wrap and place in a warm place to rise for 1 hour. They will be puffy and almost be full-size when ready.
8.Meanwhile, preheat the oven to 375 degrees. Whisk the egg yolk with 1/2 teaspoon water. Brush generously on each bun, covering the entire surface, and bake for 16-19 minutes, until deeply golden brown.