Make the dressing: whisk together the oil, peanut butter, soy sauce, lime juice, minced ginger, minced garlic, monkfruit sweetener, and water in a bowl until smooth and pourable.
Prepare the chicken: cut cooked chicken into bite-sized pieces to yield about 3 cups and place in a large salad bowl.
Toss the chicken with about half of the dressing, then let it marinate while you prepare the vegetables.
Prep the vegetables: trim and chop broccoli into bite-sized florets to make about 5 cups, and trim, seed, and dice the red bell pepper into 1/2-inch pieces.
Add the broccoli and red pepper to the bowl with the dressed chicken and stir to combine; add more dressing until the salad is as moist as you prefer.
Fold in the chopped peanuts if using, reserving some to sprinkle on top, then serve immediately or chill for a few hours.