Split the breasts open and cut them across horizontally. Pound the meat out a little then stuff with ham and cheese. Fold over and press edges to secure stuffed breasts. Season the chicken with salt and pepper, and coat in flour mixed with cornstarch, then egg white, then breadcrumbs tossed with melted butter mixed with hot sauce.
Heat a shallow layer of oil, about 1/4-inch deep, in a large skillet over medium to medium-high heat. When oil is hot, cook chicken on each side until golden and cooked through, about 6 minutes on each side. Drain chicken briefly on paper towels and serve with carrot and celery sticks alongside.