Placed crushed chips on large plate. With fork, beat egg in pie plate or shallow dish. Dip 1 chicken breast half in egg, then transfer to crumbs in plate, pressing chicken so crumbs adhere. Transfer chicken to ungreased cookie sheet. Repeat with remaining chicken.
Dip chicken again in remaining crumbs so that chicken is completely coated. Return chicken to cookie sheet. Sprinkle both sides of chicken with 1/4 teaspoon pepper. Bake just until chicken loses its pink color throughout, about 15 minutes.
Meanwhile, in large bowl, toss cabbage mix with carrot, vinegar, oil, sugar, salt, and remaining 1/8 teaspoon pepper until well combined. In small bowl, stir mayonnaise and barbeque sauce until well blended.
To serve, spoon barbeque mayonnaise into 4 small cups. Divide chicken and slaw among 4 dinner plates and serve with mayonnaise.