Combine broth, juice, ginger and garlic in saucepan. Bring to simmering. Add chicken; cook, covered, 15 minutes, until no longer pink in center; turn over halfway through cooking.
Add the snow peas; cover and cook until tender-crisp, 2 minutes. Remove chicken and snow peas; reserve liquid. Cool chicken; cut into cubes. Combine chicken and snow peas in bowl.
Reduce cooking liquid to 3/4 cup over high heat, 10 minutes. For dressing, whisk into reduced liquid the honey, peanut butter, vinegar, soy sauce and oil. Serve dressing warm, or quick-chill in freezer 15 minutes.
Line platter with cabbage and romaine. Add water chestnuts, red pepper and half of the dressing to chicken. Spoon onto cabbage-romaine mixture. Sprinkle with peanuts. Serve remaining dressing on side. Makes 4 servings.