Dressing
Whisk mayonnaise, honey, lemon juice, walnut oil, salt and pepper together until emulsified. Set aside.
Salad
Cook barley in a small pan with 2 cups of water over medium heat, partially covered, for about 20-30 minutes until tender. Season with salt and pepper.
Toast walnuts in a small pan over medium heat until visibly darker in color and fragrant, (about 3-5 minutes).
Toss apples, celery root, cherries, raisins, endive, radicchio with walnuts and barley. Season with salt and pepper. Toss with dressing.
Season arugula with walnut oil, salt and pepper. Divide among 6 plates or arrange on a platter. Arrange salad on top of greens. Garnish with dried apple slices and pomegranate arils.