Use a spiralizer to cut the zucchini into zoodles. Add the zoodles to a serving bowl and set aside.
Make the pesto: in a food processor or blender, add the spinach, basil, pine nuts, Parmesan, garlic, olive oil, lemon juice, salt, and pepper. Blend to desired consistency, adding more olive oil as needed.
Scoop the pesto over the zucchini noodles, then add the cherry tomatoes and prosciutto. Toss until the zoodles are well-coated.
Sprinkle with Parmesan and garnish with basil and lemon wedges.