Place the chicken in a large saucepan and pour over enough water to cover. Set over high heat and bring to a boil. Reduce the heat to medium and simmer until the chicken is cooked through, about 8 minutes.
Drain and, when cool enough to handle, cut the chicken into 1-inch pieces. Transfer the still warm chicken to a large bowl and add the cherries, apples, fennel, walnuts, and chives. Toss to combine.
In a small bowl, whisk together the mayonnaise, lemon juice, mustard, and lemon zest. Add to the chicken mixture and stir to coat. Season with salt and pepper.
Arrange the lettuce leaves on a serving platter or individual plates. Spoon the chicken mixture evenly into the lettuce leaves and serve. If serving chilled, you can make the chicken mixture up to 2 days in advance and store in a sealable container in the refrigerator.