To make mashed potatoes, add the 6 potatoes to a pan. Cover with water, bring to a boil, and cook until tender when pierced with a sharp knife, about 20 minutes. Drain, mash until smooth. Add milk, butter, shredded cheese and salt and pepper to taste.
Spread mashed potatoes into a greased 9 x 13 inch baking dish. Heat oven to 450°F.
Slice each of the 4 halved potatoes into 1/8 inch slices, keeping them in a line. Fan out at an angle and place in rows on the top of the mashed potatoes. Sprinkle with salt and pepper, brush with melted butter and top with parmesan cheese.
Bake for about one hour, until potatoes are tender. If desired, sprinkle with parsley, paprika or both before serving.
