Bring a large pot of salted water to a boil. Add the pasta and garlic and cook as the pasta label directs; reserve 1 cup of the cooking water, then drain.
Meanwhile, thinly slice the squash on the wide side of a box grater (or use a mandoline). Combine the squash, tomatoes, olive oil and basil in a large bowl; add 1 teaspoon salt, and pepper to taste.
Mash the cooked garlic with the flat side of a chef's knife; add it to the bowl along with the pasta, grana padano and mozzarella. Toss until combined, adding the reserved pasta water as needed to moisten. Season with salt and pepper. Divide among bowls; drizzle with olive oil and top with more grana padano.