In a large bowl, combine the egg, pickle relish, 6 tablespoons bread crumbs and mustard. Crumble ham over mixture and mix well. Gradually add milk if mixture is too thick. Shape into four patties. Sprinkle each side with remaining bread crumbs.
In a large skillet, cook patties over medium heat in butter for 4-5 minutes on each side or until a thermometer reaches 160°.