Bring a large pot of water to a boil over medium-high heat. Generously salt water and return to a boil. Add linguine and cook, according to package directions, until al dente.
Meanwhile, in a large skillet, heat 2 tablespoons olive oil over medium heat. Add onion and cabbage. Cook, stirring, until onion is soft and cabbage is wilted, about 2 minutes. Add leek, corn, and kale. Cook, stirring, until leeks are soft and kale is wilted, 2 minutes more. Season with crushed-red pepper flakes.
Add garlic, basil, and cherry tomatoes. Cook, stirring, until garlic is fragrant and tomatoes begin to release their juices, about 5 minutes.
Drain pasta, reserving 1/2 cup cooking liquid. Transfer pasta to skillet along with cooking liquid and toss to combine. Serve immediately with grated cheese and garnish with basil.
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