In medium bowl, mix hoisin sauce, soy sauce, mustard, lemon juice, sesame oil and ginger. Measure out 1/2 cup mixture; cover and refrigerate for use in step 3. Place remaining mixture in large resealable food-storage plastic bag. Add chicken; seal bag and marinate in refrigerator 4 hours.
Heat gas or charcoal grill. Remove chicken from marinade, reserve marinade. Place chicken on grill over medium heat. Cover grill; cook 25 to 30 minutes, turning frequently and brushing with marinade from plastic bag, until juice of chicken is clear when thickest piece is cut to bone (170°F for breasts; 180°F for thighs and drumsticks).
Before serving, brush chicken with the 1/2 cup mixture reserved in step 1.