Step 1: Activate the Yeast

1. In a small bowl, combine the warm milk, yeast, and 1 tablespoon of sugar. Stir gently and let it sit for 5-10 minutes until the mixture becomes frothy. This indicates the yeast is active.

Step 2: Mix the Dough

1. In the bowl of a stand mixer fitted with the dough hook, combine the flour, remaining sugar, and salt. Mix briefly to combine.
2. Add the frothy yeast mixture, eggs, and vanilla extract to the flour mixture. Mix on low speed until the dough begins to come together, about 2-3 minutes.
3. Gradually add the softened butter, one piece at a time, allowing each piece to incorporate before adding the next. This process may take 5-7 minutes.

Step 3: Knead the Dough

1. Increase the mixer speed to medium and knead the dough for 8-10 minutes. The dough should be smooth, elastic, and slightly sticky.
2. Lightly flour a clean surface and turn the dough out onto it. Shape it into a ball.

Step 4: First Rise

1. Lightly grease a large mixing bowl with oil or butter and place the dough inside. Cover it with plastic wrap or a clean kitchen towel.
2. Let the dough rise in a warm, draft-free spot for 1-2 hours, or until it has doubled in size.

Step 5: Shape the Rolls

1. Punch down the dough to release excess air and turn it out onto a lightly floured surface.
2. Divide the dough into 12 equal portions (about 70g each).
3. Roll each portion into a smooth ball, tucking the edges underneath to create a taut surface.

Step 6: Second Rise

1. Arrange the rolls on a parchment-lined baking sheet, spacing them about 2 inches apart.
2. Cover loosely with plastic wrap or a kitchen towel and let them rise for 1 hour, or until puffy and nearly doubled in size.

Step 7: Prepare the Egg Wash

1. In a small bowl, whisk together the egg yolk and milk.
2. Brush the egg wash over the surface of each roll to create a glossy, golden crust.

Step 8: Bake the Rolls

1. Preheat your oven to 375°F (190°C).
2. Bake the rolls for 20-25 minutes, or until they are golden brown and sound hollow when tapped on the bottom.
3. Remove from the oven and let the rolls cool slightly on a wire rack before serving.