Cook the pasta according to the package directions.
Meanwhile, place the garlic, pine nuts, basil, 1/2 teaspoon salt, and 1/4 teaspoon pepper in the bowl of a food processor and pulse to combine. While the machine is running, drizzle in 2/3 cup of the oil through the feed tube, stopping occasionally to scrape down the sides. Transfer to a bowl and stir in the Parmesan.
Heat the remaining 1 tablespoon olive oil in a skillet over medium heat. Add the zucchini, 1/2 teaspoon kosher salt, and 1/8 teaspoon black pepper. Cook, stirring occasionally, until tender, 4 to 5 minutes. Stir in the mint.
Divide the pasta among bowls and spoon the pesto and zucchini over the top.