Peel the potatoes, cut them into 1/2-inch cubes, rinse in cold water, and set aside.
In a soup pot, cook the butter, oil, sliced onions, and a healthy pinch of salt over medium heat, uncovered, stirring occasionally, until the onions wilt and turn a pale brown.
Add the diced potatoes, turn the heat up to high, and saute briskly, turning the potatoes through the onions to coat them well.
Add the broth, cover, and adjust heat so the soup comes to a slow, steady boil. When the potatoes are very tender, mash most of them against the side of the pot with a long wooden spoon. Stir well and cook another 8 to 10 minutes. If the soup becomes too thick, add up to a ladleful of broth (or water if you are not using homemade broth).
Before turning off the heat, stir in the Parmesan and parsley. Taste and adjust salt, then serve with additional grated cheese at the table.