Bring a large pot of salted water to a boil. Add the pasta and cook until just short of al dente, 8 minutes. Drain, reserving 1 1/4 cups of the pasta water.
Meanwhile, heat 1/4 cup of the olive oil in a large skillet over medium-high heat. Add the chickpeas and cook, stirring often with a wooden spoon, until golden brown and crispy, about 5 minutes. Add the garlic, chile flakes and capers and cook until fragrant, 1 minute. Stir in the Swiss chard and 1/4 teaspoon salt; cook until wilted, 3 minutes. Add 1/4 cup of the reserved pasta water and simmer for 1 minute. Add the drained pasta and sprinkle with 1 1/2 cups of the Parmesan and the lemon zest; toss well to coat. Add 1 cup more of the reserved pasta water and the remaining 1/4 cup olive oil. Simmer until the pasta is al dente, the sauce has thickened and the flavors have come together, 2 minutes. Top with the remaining 1/4 cup Parmesan and serve.