Line a baking sheet or your kitchen counter with a cotton towel or paper towels. Carefully remove each whole bibb lettuce piece, starting from the outside and working your way in. Rinse each piece of lettuce under cool running water to remove any dirt or grit. Shake off excess water and place on the towels to air dry while you prepare the chicken salad.
Shred the chicken and place in a large salad or mixing bowl. Thinly sliced the celery, carrot, fennel, radish, and green onion using a mandoline slicer or very sharp knife and add them to the chicken, along with the dried cherries and almonds.
In a small bowl, combine the vinegar, balsamic reduction, parsley, salt, and pepper and whisk until combined. Slowly whisk in the olive oil until the vinaigrette is emulsified.
Pour the balsamic vinaigrette over the chicken salad and toss until combined.
Spoon 1/12th of the chicken salad in 12 individual lettuce cups. Pack two cups in a sealed container and store in the fridge for up to four days, or arrange on a platter and serve immediately. Alternatively, store the chicken salad and lettuce cups in separate sealed containers to keep in the fridge for up to a week. This chicken salad can be enjoyed immediately, but the flavor gets better after marinating for at least an hour.
Notes
A portion sized represents two lettuce wraps, each stuffed with 1/12 of the cherry chickens salad. "Cooking time" represents assembly time to mix the chicken salad and fill the lettuce wraps.