Do Ahead: Slaw can be made 1 day ahead. Cover and chill.
Toss red onion and a pinch of salt in a small bowl to coat; let sit 5 minutes to allow onion to soften slightly. Rinse under running water, then shake off excess water. Place onion in a medium bowl and add cabbage, vinegar, and 2 Tbsp. oil. Toss to combine and season slaw with salt and pepper.
Pound chicken thighs with a meat mallet or a rolling pin between 2 pieces of plastic wrap to ¼" thick.
Place cornstarch in a shallow bowl. Whisk eggs, mustard, and cayenne in another shallow bowl. Place panko in a third shallow bowl. Working with 2 thighs at a time, season chicken with salt and pepper. Dredge in cornstarch, shaking off excess. Dip in egg mixture, letting excess drip back into bowl, then coat in panko, shaking off excess.
Heat remaining 2 Tbsp. oil in a large skillet over medium-high. Working in batches and adding more oil to skillet between batches if needed, cook chicken thighs until golden brown and cooked through, about 2 minutes per side. Transfer to paper towels to drain.
Spread mayo over one side of bread slices and build sandwiches with cutlets and slaw.