In the bottom of a salad bowl, whisk together the shallot, vinegar, 1 teaspoon salt, and ½ teaspoon pepper. Drizzle in the olive oil while whisking. (Dressing can be prepared in advance. Give it another whisk before adding the rest of the ingredients and tossing just before serving time.)

Add the apples, walnuts, and arugula to the bowl and toss well. Adjust seasoning with more salt and pepper if desired.