In a food processor, combine the arugula, basil, mint, 3 tablespoons walnuts, garlic, lemon zest and juice, 1/2 teaspoon salt and 1/4 teaspoon pepper. Pulse until coarsely chopped. With the machine running, drizzle in the olive oil; process until smooth. Store in the refrigerator for up to 3 days.
Using a vegetable peeler, cut the zucchini into lengthwise ribbons. Add to a large bowl and toss with the 1/3 cup walnuts and the goat cheese. Add the dressing and toss to combine.