Preheat your oven to 400°F. Lightly grease a shallow enameled cast iron braiser with 1 tablespoon of salted butter.
To a saucepan, add the remaining 3 tablespoons butter, cream, milk, chopped herbs, Kosher salt, and black pepper. Bring to a simmer, stirring constantly, until the butter is melted and the mixture is heated through. Cut the heat but leave atop the warm burner.
Begin a layering process using overlapping sweet potato rounds in the bottom of the braiser. Make sure the entire surface of the cast iron bottom is covered. Next, sprinkle with 1/3 of the nutmeg followed by 1/3 of the Muenster and 1/3 of the warm cream mixture. In the skillet, arrange the potatoes in an even layer.
Repeat with the remaining potatoes, nutmeg, Muenster cheese, and cream mixture to form 3 layers. Press down on the layers to completely submerge the sweet potatoes in the cream mixture and top with additional cheese and a very small final grind of nutmeg.
Cover and bake for 30 minutes. Remove the cover and continue baking until the cream has been absorbed, the potatoes are cooked through, and the top is browned, another 30 to 45 minutes.
Remove from the oven. Allow the gratin to stand for 15 minutes before serving. Sprinkle with additional fresh minced herbs if desired.