Heat oil in a skillet over moderately high heat until hot but not smoking, add potatoes and onion and cook for 5 to 7 minutes, shallots are caramelized and potatoes start to brown.
Meanwhile, bring a large pot of water to a boil and blanch kale for 5 minutes. Remove kale with a large sieve and drain. Run under cold water and then roughly chop. Return cooking water in pot to a boil, then cook pasta in boiling water, uncovered, until al dente. Reserve 1 cup pasta-cooking water, then drain pasta in a colander.
While pasta cooks, add kale to potatoes in skillet and sauté, stirring frequently, until just tender, about 5 minutes. Add broth, stirring and scraping up any brown bits from bottom of skillet, then add pasta and 1/2 cup reserved cooking water to skillet. Toss until well combined. Divide pasta between 2 bowls and sprinkle basil and red pepper flakes.