Heat grill. In a bowl, combine all chicken ingredients; toss together to coat chicken completely. Grill, turning once, until chicken is cooked through, about 4 minutes per side.
Pour slaw mix into a large bowl. In a small saucepan, combine vinegar, sugar, vegetable oil, hot sauce, and salt. Bring to a boil, pour over slaw mix, and toss to combine. Add cucumber and mint, and toss again. Top with grilled chicken. Serve chow-mein noodles on the side.