In a small bowl, stir lukewarm milk with ⅓ tsp sugar and the yeast. Set aside 5–10 min until foamy.
In a stand mixer bowl add the yeast mixture, sugar, salt, whole eggs, and extra yolk.
Gradually add the flour and knead lightly.
On low speed, add softened butter 2 tbsp at a time, waiting until each piece disappears before adding the next. If dough is overly sticky after butter is incorporated, add up to 2 tbsp more flour.
Increase to medium‑low and knead 8–10 min. The dough should pull away from the bowl and feel smooth, elastic, and slightly tacky.
Lightly oil a large bowl. Shape dough into a ball, place inside, turn to coat, and cover. Let rise in a warm spot until doubled, 1½–2 hours.
Punch down, then divide into 10 equal pieces (about 70 g each). Cup your hand and roll each piece on an unfloured surface to form tight balls. Arrange on a parchment‑lined baking sheet, leaving space to expand.
Cover with a damp towel or plastic wrap; proof at room temp until doubled, 1 to 2 hours, depending on room temperature.
Preheat oven to 350 °F / 180 °C. Whisk 1 egg with a pinch of salt; very gently brush tops. Bake 22–25 min until deep golden and buns sound hollow when tapped.
Transfer to a wire rack; cool completely for best texture (crumb sets as they cool).