Mix egg yolks, mashed potatoes, Parmesan, chives, and flour in a medium bowl. Roll into walnut-size balls; chill until cold, at least 2 hours.
Beat eggs in a bowl; place breadcrumbs in another bowl. Pour vegetable oil into a medium skillet to measure ½” (about 2 cups) and heat over medium-high until a pinch of breadcrumbs bubbles immediately when added. Dip potato balls in egg, then roll in breadcrumbs. Working in batches, fry, turning often, until golden brown and crisp, 3–4 minutes. Drain on paper towels.