Place chicken breasts between two sheets of plastic wrap and pound to an even thickness of about 1/2 inch. Season with salt and pepper.
Set up a breading station: flour in a shallow dish, beaten eggs in another, and panko mixed with a pinch of salt and pepper in a third.
Dredge each chicken breast in flour, shaking off excess. Dip into egg, then coat with panko, pressing gently to adhere.
Heat vegetable oil in a large skillet over medium-high heat. Fry chicken for 3-4 minutes per side until golden brown and cooked through (internal temp 165°F). Transfer to a paper towel-lined plate.
In a bowl, combine shredded cabbage, carrots, mayonnaise, rice vinegar, and sugar. Toss to make slaw.
Warm tortillas in a dry skillet or microwave for about 15 seconds.
Slice each chicken cutlet into strips. Lay a tortilla flat, place a portion of slaw in the center, top with chicken strips, and drizzle with katsu sauce.
Fold in the sides of the tortilla, then roll tightly from the bottom to form a wrap. Cut in half diagonally and serve.
Tips
For extra crunch, toast the panko in a dry pan before breading.
Katsu sauce can be made by mixing ketchup, Worcestershire sauce, soy sauce, and sugar to taste.
Wraps can be assembled ahead and refrigerated; reheat in a panini press or skillet to crisp.