In large resealable food-storage plastic bag, place 1 cup of the dressing and the chicken; turn bag to coat. Refrigerate at least 30 minutes to marinate.
Heat gas or charcoal grill for medium heat.
Place chicken on grill; discard marinade. Cover grill; reduce heat to low and cook 7 minutes. Turn chicken; cook 7 to 8 minutes longer or until juice of chicken is clear when center of thickest part is cut (170°F). Cut chicken across grain into thin strips.
In 7-quart bowl, gently toss chicken, salad greens, strawberries and remaining 3/4 cup dressing. Sprinkle with peanuts.