Preheat the oven to 400°F (200°C). Place an oven-safe wire rack on a rimmed baking sheet and set aside; line the sheet with parchment if you prefer not to use a rack.
Halve each chicken breast lengthwise to make two thin cutlets (about 4 total). If pieces are uneven, gently pound the thicker parts to an even thickness for uniform cooking.
Season both sides of each chicken piece with the kosher salt and black pepper.
Place the flour in one shallow dish and the panko in a second shallow dish. In a bowl, whisk the egg with 1 tablespoon water until combined.
Working one piece at a time, dredge the chicken in flour, shake off excess, dip into the egg wash, then press firmly into the panko so breadcrumbs adhere to both sides.
Arrange the breaded chicken on the prepared rack or parchment-lined sheet. Lightly spray both sides of each piece with avocado oil spray.
Bake for 20–25 minutes, flipping once halfway through if not using a rack, until the internal temperature reaches 165°F (74°C) and the crust is golden and crisp.
Meanwhile, whisk together ketchup, Worcestershire sauce, soy sauce, mirin, granulated sugar, and Dijon mustard (if using) in a small bowl until smooth.
Serve the chicken katsu hot with the sauce on the side or spooned over the cutlets for serving.
