Add the ground beef to the onions. Season with 1/4 teaspoon salt and 1/8 teaspoon black pepper. Break up the meat with a wooden spoon and cook until no pink remains.
Fold the cooked meat and onions into the mashed potatoes. Mix gently until combined.
Set up three bowls: one with flour, one with beaten eggs, and one with panko breadcrumbs. Dredge each chilled patty in flour, dip in the egg, and then coat evenly with panko.