The original Parker House rolls were shaped a particular way, where the dough was flattened, brushed with butter and folded with that butter inside. This version opts for a simpler rounded shape—and that buttery interior is swapped with a butter-basted exterior, salty Parmesan and a punch of fresh herbs. They boast the same golden-brown exterior and soft, fluffy interior that makes them perfect alongside any meal.
In a medium saucepan, heat the milk and butter over medium until the butter is melted and the milk is warm to the touch. If it feels hot to the touch, not just warm, let it cool for a few minutes before proceeding. Whisk in the buttermilk and eggs.
In the bowl of a stand mixer fitted with a dough hook, mix the flour, Parmesan, sugar, parsley, oregano, rosemary, thyme, yeast and salt to combine. Add the milk mixture to the bowl and mix on low speed for 4 minutes, then on medium speed for 3 minutes more.
Spray a medium bowl with nonstick spray, place the dough inside and cover the top of the bowl tightly with plastic wrap. Let rise until doubled in size, 1 to 2 hours.
Spray a 9-by-13-inch baking pan with nonstick spray. Divide the dough into 12 even pieces. Roll each piece into a tight round and place in the pan. Cover with lightly greased plastic wrap and let rise until puffy, 1 to 1 1/2 hours.
Towards the end of rise time, preheat the oven to 375 degrees F. For finishing, brush the rolls with about a third of the melted butter. Bake for 15 minutes, then sprinkle the top of each roll with Parmesan. Return to the oven and bake for 10 minutes more, then brush with the remaining butter and bake until the rolls are deep golden brown and no longer doughy inside, about 5 minutes more. Cool at least 10 minutes before serving.