Warm the milk: In a microwave-safe bowl, warm the milk in the microwave for about 1 minute, until it’s between 98°F to 105°F.
Wet Ingredients: In a large mixing bowl, whisk the warm milk with the sugar, instant yeast, softened butter, one whole egg and one egg yolk. (Save the egg white in a small bowl in the fridge for the egg wash topping later.)
Dry Ingredients: Add the flour, garlic powder, onion powder, Italian seasoning, and salt. Mix with a large wooden spoon until it forms a shaggy dough.
Knead: On a lightly floured work surface, knead the dough by hand for about 5 minutes, until the dough is smooth and elastic. (Or knead the dough in a stand mixer with a hook attachment for about 3 minutes.)
Rise: Place the dough ball back in the mixing bowl and cover it with a clean kitchen towel. Let the dough rise in a warm spot until it has doubled in size, after about 1 hour.
Divide: Divide the dough into 16 equal-sized pieces, and keep them covered with a kitchen towel so the dough doesn’t dry out.
Shape and Fill: Line a baking sheet with parchment paper. Roll out one piece of dough into a flat circle with a rolling pin, then add a handful of shredded cheese on top in the center. Fold the edges of the dough over the cheese and pinch the edges together, so the cheese is fully enveloped in the dough. Place the dough seam-side-down in the prepared baking sheet. Repeat with the remaining dough & cheese.
Proof: Cover the dough loosely with a clean kitchen towel. Set aside to proof for 30 minutes, until the dough balls have puffed up.
Egg Wash: Preheat the oven to 350°F. Whisk the egg white with ½ teaspoon of water in a small ball to make an egg wash. Once the rolls are done proofing, brush the tops of each roll with the egg wash.
Bake: Bake the bread for 20 to 25 minutes, or until the rolls are golden on top.
Toppings: Brush the tops and sides of the buns with the melted butter and sprinkle with fresh herbs.