Make the tartar. Mince the onion, soak in water and squeeze dry (for pickled rakkyo, just chop and drain lightly, no soaking; its sweet-vinegar tang means you can skip the lemon juice). Roughly mash the boiled eggs and mix with the onion, mayonnaise, lemon juice, salt, pepper and chopped parsley.
Bring the nanban sauce ingredients to a brief boil in a small pot to dissolve the sugar (30 seconds in the microwave also works).
Open out the thick part of the chicken to even the thickness and cut in two. Season with salt and pepper and dust thinly all over with cornstarch.
Heat oil in a pan over medium, dip the chicken in beaten egg and lay it in skin-side down. Shallow-fry for 4-5 minutes, flip once browned, cover and cook over medium-low for 4 minutes until cooked through.
Wipe off excess oil, add the nanban sauce and coat both sides.
Cut into easy-to-eat pieces, plate, top with plenty of tartar and serve with cabbage and cherry tomatoes.