Zest one lemon to measure 1 tablespoon of lemon zest, then squeeze the lemon to yield 1/4 cup fresh lemon juice.
In a mixing bowl whisk together lemon zest, lemon juice, 2 tablespoons spicy brown mustard, 1/2 cup extra-virgin olive oil, and fresh-ground black pepper to make the dressing.
Place the cooked chicken pieces in a large bowl and toss with 1/2 cup of the dressing to lightly coat; let the chicken marinate while you prepare the remaining ingredients.
Drain the hearts of palm and pat them dry. Cut into thick slices. Drain olives in the same colander and slice the green onions.
Gently combine the marinated chicken, sliced hearts of palm, drained olives, and sliced green onions in the bowl.
Add more dressing as needed and toss until the salad is moistened to your liking; season with salt and additional fresh-ground black pepper to taste.
Serve immediately or chill for a few hours; if chilled, stir before serving and add a little extra dressing if desired.