In a small pot, simmer the butter and half-and-half just until the butter melts. (Do not boil.) Set the pot aside.
Layering process / repeat three times: potatoes, salt and pepper, herbs, thin layers of the cheeses and a dusting of Parmigiano Reggiano.
After you’re finished with the three layers, pour the half-and-half mixture evenly over the casserole. Add a thin layer of Parmigiano Reggiano on top.
Cover the pan loosely with foil (so the cheese doesn’t touch) and bake for 30 minutes. Remove the foil, lower the oven to 350° and bake for an additional 30 minutes. The top should be a deep golden brown.