In a small bowl or jar, combine olive oil, lemon juice, a generous pinch of salt, and a pinch of pepper, and whisk to combine. Taste and adjust seasoning as needed, and then set aside.
Heat a small pan to medium-high heat, add the chopped walnuts to the hot, dry pan, and cook, stirring them frequently until lightly browned and toasted, then remove from the heat and set aside.
Using a sharp knife, cut the root off the fennel bulbs and remove the fronds, setting them aside. Using a mandoline or knife, slice the fennel bulbs as thinly as possible, and then roughly chop the fronds with a knife.
Using a sharp knife, cut the apple 1/4 to 1/2 inch away from the center stem, then turn 90 degrees until you have cut a total of 4 pieces and only the core remains. Dispose of the apple core and, using a mandoline or knife, slice the apple pieces into thin slices.
In a large bowl, add arugula, fennel, apple, and fennel fronds, and toss to combine. Drizzle with the dressing and toss again until everything is well coated. Taste and taste salt and pepper seasoning as needed.
Top the salad with toasted walnuts and, using a vegetable peeler, shave the pecorino or parmesan cheese over the salad.