Zest one of the limes with a rasp grater and cut into 6 wedges. Cut the second lime into 6 wedges. Transfer the zest into a large bowl and set the wedges aside.
Add the egg, mayonnaise, kimchi, pepper, and salt to the bowl with the lime zest. Whisk to combine. Fold in 43 grams (roughly half) of the panko and the crab meat.
Fill a pie pan with the remaining panko.
Divide the mixture into 12 (2-ounce) portions and shape into patties. Transfer the cakes to the pie pan, coat both sides in panko, then place on a cooling rack set inside a half-sheet pan. Refrigerate for 30 minutes.
Heat the oil in a 12-inch saute pan over medium heat. When the oil shimmers, cook the crab cakes, 4 at a time, for approximately 3 minutes per side. Remove to a clean wire rack set over paper towels to drain. Repeat with remaining crab cakes.
Serve immediately with lime wedges.