Trim the bottom and green top from the fennel bulb. Shave the bulb into paper-thin slices, preferably using a mandoline. Remove the core from the crisp apple and slice the flesh into thin slices. Finally, slice the smoked chicken breast into even pieces.
Heat a frying pan over medium heat without adding oil or butter. Toast the walnuts briefly until they begin to color and smell fragrant. Remove the nuts from the hot pan immediately, let them cool briefly, and coarsely chop with a knife.
Whisk the olive oil and lemon juice together in a small bowl. Whisk vigorously until you have a smooth dressing.
Arrange the arugula on the bottom of a large serving platter. Arrange the fennel, apple, and smoked chicken slices evenly over the greens. Drizzle the dressing over the salad and finally sprinkle the toasted walnuts on top.