Using potato masher, mash potatoes and ham in medium bowl. Stir in 2 tablespoons breadcrumbs and cheese. Roll mixture into 1-inch balls.
Beat eggs in shallow bowl. Place remaining breadcrumbs in another shallow bowl. Dip each potato ball into egg; then roll into breadcrumbs, completely coating. Set aside.
Heat oil in large skillet over medium-high heat. Carefully place potato balls in oil and fry, turning frequently until lightly browned and crispy, 3 to 5 minutes. Remove with slotted spoon and place on paper towel-lined plate to drain. Serve hot, with sour cream on the side.