Heat olive oil in a large skillet on medium-high heat. Add zucchini, corn, onion, and a pinch of salt and saute 5-7 minutes until onion is translucent and zucchini is crisp-tender. Add cherry tomatoes, garlic, and red pepper flakes and saute until vegetables are lightly browned and tender. Season with salt and black pepper to taste.
While vegetables are cooking, bring a large pot of salted water to a boil and cook gnocchi to package directions. Drain and add to skillet with the vegetables.
Make the brown butter. Heat butter and basil in a skillet (preferably with a light color so you can see how the butter is browning) on low heat. Cook until melted. Continue to cook as butter spatters, then eventually starts to foam, stirring regularly to scrape up the bottom and prevent burning. Cook until milk solids are golden brown, about 7 minutes total. Here's a link with pictures if you're not sure. Scrape butter over the pasta and toss to combine. Serve garnished with parmesan.