Mix ham, mashed potatoes, green onions, egg, Dijon mustard, dill weed, salt, black pepper, and 1/4 cup panko bread crumbs together in a bowl until thoroughly combined. If mixture is too moist to hold together, mix in an additional 1/4 cup of crumbs.
Place remaining panko crumbs into a shallow bowl.
Scoop about 1/3 cup of the mixture and form into a ball; slightly flatten into a patty. Press both sides of patty into the panko crumbs; tap off any loose crumbs. Repeat with remaining mixture.
Heat vegetable oil in a skillet over medium heat until very hot; pan-fry 2 or 3 patties at a time in the hot oil until golden brown on each side, 3 to 4 minutes per side.
