Bring a large pot of salted water to a boil. Add the pasta and cook until al dente, about 8 minutes. Drain well.
Meanwhile, in a large skillet, heat the oil over medium-high heat. Add the bell peppers and onions, and cook, stirring, until soft, about 4 minutes. Add the garlic, red pepper flakes, salt and pepper, and cook for 1 minute. Add the wine and stir to deglaze the pan. Add the vegetable stock and chickpeas and bring to a boil. Reduce the heat to low and simmer until the chickpeas are tender, about 5 minutes, and almost all the broth is gone. Crush the peas lightly with the back of a spoon against the side of the pot. Add the spinach and cook until wilted, about 2 minutes. Add the cooked pasta and toss well to coat. Adjust seasoning, to taste. Add the extra-virgin olive oil, cream, and cheeses, and toss to combine.
Divide the pasta among serving plates and serve immediately.
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