Crush or finely chop the garlic. Thinly slice the spring onion keeping the white and green parts separate. Shred the cabbage with a sharp knife or mandoline. Thinly slice half the unpeeled apple, then peel the remaining apple and cut into 2cm chunks.
Step 2 — Make pork patties
Put the pork mince, garlic, white part of the spring onion, sesame oil, 2 tbs breadcrumbs, 1 egg yolk and ½ tsp salt in a large bowl and season with pepper. Using damp hands, combine well, then shape into 2 equal patties, slightly larger than the base of a bun.
Step 3 — Coat patties
Whisk 1 egg in a bowl. Combine 2 tbs flour and a pinch of salt in a shallow dish. Combine the sesame seeds and remaining breadcrumbs in a separate shallow dish. Dust the patties with the flour, shake off the excess, then dip in the egg and coat in the breadcrumb mixture.
Step 4 — Make apple sauce
Heat 20g butter in a small saucepan over medium heat. Cook the chopped apple, covered, stirring occasionally, for 4-5 mins until golden and softened. Remove from the heat, cool slightly, then mash until almost smooth. Stir in the tonkatsu sauce and ½ tsp honey. Taste, then season with salt and pepper.
Step 5 — Fry patties
While the apple sauce is cooking, preheat the grill to high. Heat 1cm oil in a medium frypan over medium-high heat. Cook the patties for 3 mins each side or until golden and cooked through. Remove from the pan. Halve the buns through the middle and put on an oven tray, cut-side up. Grill the buns for 3-4 mins until lightly toasted.
Step 6 — Make slaw and serve
Combine the aioli and 1½ tsp white vinegar in a large bowl and season with salt and pepper. Add the cabbage, sliced apple and remaining spring onion to the aioli and toss to combine. Spread the buns with the apple sauce. Fill with the patties and some of the slaw. Serve the katsu burgers with the remaining cabbage slaw.