Combine the warm milk in a bowl with active dry yeast. Let it sit for 5 minutes until it becomes foamy on top.
Pour the yeast mixture into your stand mixer bowl. Add 4 tablespoons of melted butter and 2 beaten eggs. Mix on low speed for about 1 minute until combined.
Add 3 cups bread flour and 1 teaspoon salt. Mix on low speed until the dough looks shaggy.
Switch the mixer to medium speed and knead the dough for 8 minutes. The dough should become smooth and slightly sticky.
On low speed, mix in 3 minced garlic cloves and ½ cup grated Parmesan cheese until evenly distributed (about 1 minute).
Transfer the dough to a lightly oiled bowl and cover with a clean kitchen towel or plastic wrap. Let it sit in a warm spot for 1 hour until doubled in size.
Punch down the dough gently to release air.
Divide the dough into 10 equal pieces. Roll each piece into a ball.
Arrange the balls close together in a greased 9 x 13-inch baking pan.
Cover the pan loosely with a towel and let rise for 45 minutes until puffy and touching.
Mix together the dried garlic, dried oregano, ½ teaspoon salt, and 3 tablespoons grated Parmesan in a small bowl.
Preheat the oven to 375 F.
Brush the tops of rolls with 3 tablespoons melted butter, then sprinkle the garlic-herb-Parmesan mixture evenly over all the rolls.
Mix together ¼ cup softened butter, 2 minced garlic cloves, and a pinch of salt and pepper.
Bake the rolls for 16-18 minutes until golden brown on top and serve them with garlic butter on the side for dipping or spreading on warm rolls.