If planning to bake immediately, preheat oven to 350°F. Butter an 8 x8-inch or a 9 x9-inch casserole with 1 tablespoon of the butter.
Peel and quarter the potatoes and add them to a large stock pot. Add enough COLD tap water to cover by 1/2-inch. Sprinkle with Kosher salt, about 1/2 tablespoon.
Bring the pot to a boil and boil the potatoes for about 15 minutes or until a fork will stab them without resistance.
Melt the butter. Using a hand mixer or a potato masher, add to the potatoes and mash coarsely.
Fold the chopped chives, dry-cured ham or prosciutto if using, and the green parts of the scallions into the mashed potatoes, along with the shredded cheese.