Combine first 4 ingredients in a small saucepan over medium heat. Stir in sugar. Cook for 10 minutes or until sugar dissolves, stirring occasionally. Remove from heat. (Mixture will thicken.) Reserve 1 1/2 cups soy mixture, and set aside.
Brush both sides of chicken evenly with remaining soy mixture; cover and let stand 10 minutes.
Grill, covered with grill lid, over medium-high heat (350° to 400°) 6 to 8 minutes on each side or until done.
Toss together broccoli slaw, green onions, sesame seeds, and 1/2 cup reserved soy mixture; top with grilled chicken. Season with salt and pepper to taste. Serve with remaining reserved 1 cup soy mixture for dipping.
Grilled Salmon With Sweet Soy Slaw and Dipping Sauce: Substitute 6 (4-ounce) salmon fillets for chicken, and grill 4 to 6 minutes on each side or until fish flakes with a fork.