Spiralize the zucchini into noodles using the medium blade on your spiralizer. Spread on a clean kitchen towel, sprinkle with kosher salt, and let sit 10 minutes to draw out moisture. Gently squeeze dry with the towel. This prevents watery zoodles.

Heat 2 tablespoons olive oil in a large skillet over medium-high heat until shimmering. Add the cherry tomatoes and corn. Cook undisturbed for 2 minutes until the tomatoes blister and the corn chars in spots. Stir once and cook 1 minute more.

Push the vegetables to the sides of the pan. Add the remaining 1 tablespoon olive oil to the center, then add the garlic and red pepper flakes. Saute 30 seconds until fragrant but not browned.

Add the zoodles to the skillet. Toss everything together with tongs and cook for 2 to 3 minutes, just until the zoodles are tender but still have a slight bite. They will continue cooking from residual heat, so err on the side of underdone.

Remove from heat. Immediately add the lemon juice, grated Parmesan, and torn basil. Toss vigorously so the cheese melts slightly and coats the zoodles. Taste and adjust seasoning. Divide among plates, top with more Parmesan and flaky salt, and serve right away.