Trim the stem end off each zucchini. Use a mandolin (sold at well-stocked kitchenware stores) to cut each zucchini, lengthwise, into julienne slices (long strips). If you don’t have a mandolin, use a sharp knife or wide vegetable peeler to cut each zucchini into long, thin slices. Now cut each of those slices, lengthwise, into ¼" wide strips. Set the zucchini in a bowl.
Heat 2 Tbsp. of the oil in a large skillet set over medium heat. Add the garlic, anchovies and chilies and cook until fragrant, about 1 minute. Mix in tomato sauce and paste, stock, olives and capers. Bring to a simmer, simmer 2 to 3 minutes, and then cover and reserve on low heat.
Place the remaining oil in a large, wide skillet or pot set over medium, medium-high heat. Add the zucchini and water and cook and stir the zucchini until just softened and flexible, about 5 minutes.
Divide the zucchini among four plates or bowls. Top with sauce, sprinkle with cheese and basil, and serve.