Begin heating the frying oil to 350°F in a large, high-sided skillet while you fold the chicken, ham, 1/2 cup panko, one lightly beaten egg, salt, and pepper together in a large bowl. Stop mixing as soon as the ingredients look evenly distributed so the chicken mixture stays soft rather than tightly packed.
Roll the mixture into balls about 1.5 inches across, using light pressure. Aim for similar sizes so the centers cook at roughly the same pace.
Whisk the remaining two eggs with 1 tablespoon water in a small bowl, then combine the remaining panko and Parmesan in another bowl beside the stove. Keep the paper-towel-lined plate within reach so the breaded meatballs can move neatly from frying to draining.
Check that the oil has reached 350°F, then dip each meatball into the egg wash and cover it with the Parmesan-panko mixture before carefully lowering it into the skillet. A test breadcrumb should sizzle steadily rather than darken immediately, though this cue does not replace the thermometer.
Fry in batches for about 5 to 7 minutes, rotating the meatballs occasionally, then lift them onto the prepared plate once the chicken is cooked through. Look for an evenly golden crust, but check doneness rather than relying on the browned coating alone.
Set the oven to broil, place the drained meatballs on a baking sheet, and add 1 to 2 small Swiss squares to each before broiling for 1 to 2 minutes. Remove the sheet when the cheese melts and bubbles, staying beside the oven because the exposed edges can scorch quickly.