Preheat oven to 400 degrees F. Peel potatoes (if desired) and place in large sauce pot and cover with cold water by 1 inch. Bring to a boil and reduce to simmer, cooking for 30-35 minutes or until potatoes are tender-soft.
Meanwhile, in skillet over medium-high heat, melt 2 tablespoons butter and cook onion for 10-12 minutes or until soft and translucent. Drain potatoes thoroughly and place in large bowl with remaining butter, salt and pepper. With mixer or potato masher, blend potatoes and butter until mixed while retaining a slightly chunky texture. Fold in cooked onion, eggs, ham and chives.
Line baking sheet with parchment or lightly grease. In ring mold or biscuit cutter, sprinkle bottom interior with bread crumbs and fill mold with potato mixture until top is slightly rounded, topping again with bread crumbs. Remove ring and repeat with remaining potato mixture, spacing cakes about 1 inch apart. Bake for 20-30 minutes or until tops are golden brown and beginning to crisp. Top with Parmesan cheese and serve hot.