In a large pot of boiling, salted water, cook the pasta until al dente. Drain, reserving 1 cup of the pasta cooking water and the pot.
Meanwhile, in a small skillet, heat 2 tablespoons olive oil over medium heat. Add half of the garlic and cook until golden, about 30 seconds. Stir in the breadcrumbs and remove from the heat.
In the reserved pasta pot, heat the remaining 2 tablespoons olive oil over medium heat. Add the remaining garlic and cook until golden, about 30 seconds; add the zucchini and cook, turning occasionally, until crisp-tender, about 5 minutes. Stir in the corn, cooked pasta and reserved 1 cup pasta cooking water. Add 1 cup cheese and toss; season with salt and pepper.
Stir the remaining cheese and the basil into the breadcrumb mixture and sprinkle on top of the pasta.