Halve chicken breasts horizontally and season with 1/2 tsp. salt and 1/4 tsp. pepper.
Put flour, eggs, and panko separately into 3 wide, medium bowls. Mix 1/2 tsp. salt with flour. Beat eggs well. For each cutlet, dredge first in flour, then dip into egg, letting excess drip off. Lastly, coat both sides well with panko. Arrange cutlets on a cutting board in a single layer and set aside.
In a bowl, whisk ketchup, mayonnaise, Worcestershire, mustard, sugar, sesame oil, and garlic with remaining 1/2 tsp. salt and 1/4 tsp. pepper. Add carrot, cabbage, and green onions to dressing; toss well.
Heat vegetable oil in a large frying pan (not nonstick) over medium-high heat until just beginning to smoke. Add cutlets and brown on both sides, about 5 minutes total. Drain on paper towels.
Serve cutlets with tonkatsu slaw and lemon wedges.
Wine Pairing: Ponzi 2014 Pinot Blanc (Willamette Valley; $20).