Whisk together shallot, lime juice, lemon juice, honey, mustard, salt, and pepper; let stand for 2 minutes. Add oil in a slow, steady stream, whisking until blended. Toss together arugula, lettuce, apple, fennel, and walnuts in a large bowl.
Add dressing to salad and toss. Divide salad evenly among 4 plates and serve.
Adapted from Valerie’s Home Cooking by Valerie Bertinelli. Copyright © 2017 Oxmoor House. Reprinted with permission from Time Inc. Books, a division of Time Inc. New York, NY. All rights reserved.