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Combine the dressing, arugula and basil in a small bowl; reserve.
Using a spiral vegetable slicer, cut the zucchini and carrot lengthwise into long, thin strands. Heat 1 tablespoon of the oil in a large nonstick skillet over medium-high. Add the zucchini and carrot strands and cook, tossing often, until slightly softened but still crunchy, about 2 minutes. Remove from the heat, transfer to a colander and let drain 5 minutes.
Heat the remaining 2 tablespoons oil over medium-high. Add the onion and garlic and cook until slightly softened, 2–3 minutes. Add the tomatoes and cook 1 minute. Stir in the drained zucchini and carrot noodles and cook, tossing, until heated through, about 1–2 minutes. Remove from the heat and stir in the dressing mixture. Divide among 4 plates and sprinkle each serving with cheese.