Preheat the oven to 350 degrees F (180 degrees C).
Bring a pot of salted water with the olive oil to a boil. Then add the potatoes; cook until tender, about 15 minutes. Reserve 1 cup water, and drain potatoes. Transfer to a mixing bowl.
Mash the potatoes, adding some reserved water to soften and fluff potatoes. You will not need the whole cup of water.
Fold in parsley, Parmigiano, and scarmozza. Season with salt.
Transfer to an 8x8-inch baking dish and spread evenly. Sprinkle with breadcrumbs, grate a little Parmigiano over breadcrumbs, and scatter butter pieces on top.
Bake in the preheated oven until the top is golden, about 20 minutes.