Preheat the oven to 200 degrees F. When the oven reaches 200 degrees, turn it off. This will be a warm environment in which to let the dough rise.
Lightly oil a large bowl.
Whisk the milk, sugar, and yeast in a liquid measuring cup until the yeast dissolves. Then whisk in the egg and egg yolk.
In the bowl of a stand mixer fitted with the dough hook, mix the flour and salt until combined. With the mixer on low speed, add the milk mixture in a steady stream. Mix until the dough begins to form, about 1 minute.
Increase the mixer speed to medium and add the 8 tablespoons of softened butter, one piece at a time, until incorporated. Continue to mix until the dough is smooth and comes away from the sides of the bowl, about 8 minutes.
Turn the dough out onto a lightly floured surface and knead briefly to form a smooth ball. Transfer the dough to the oiled bowl and turn to coat. Cover with plastic wrap and place the bowl in the turned-off oven until the dough has doubled in size, about 45 minutes.
To shape the rolls: Brush a 12-cup muffin tin with 1 tablespoon of melted butter, or spray with cooking spray. Punch down the dough on a lightly floured work surface. Divide the dough into thirds. Roll each third into an 18-inch long rope. Then cut each rope into 12 equal pieces and cover the pieces with plastic wrap (there will be 36 pieces of dough).
Roll each piece of dough into a smooth ball. Place 3 balls, seam side down, in each muffin cup. Cover the muffin tin loosely with plastic wrap and let rest in the turned-off oven until doubled in size, about 20 minutes.
Remove the rolls from the oven and discard the plastic wrap. Preheat the oven to 375 degrees. Bake the rolls until golden brown, about 15 minutes, rotating halfway through baking. Brush the rolls with the remaining 1 tablespoon of melted butter. Cool in the muffin tin for 5 minutes, and then transfer to a wire rack. Serve warm.