Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, 8 to 10 minutes. Drain and reserve 1 cup of the pasta water.
In a large skillet, heat 1 tablespoon oil over medium-high heat. Add the pancetta and cook, stirring frequently, until golden and crisp, 6 to 8 minutes. Using a slotted spoon, remove the pancetta and drain on a paper towel-lined plate. Add the shallots to the pan and cook until soft, 2 to 3 minutes. Add the broth and scrape up the brown bits that cling to the bottom of the pan with a wooden spoon. Stir in the snap peas and simmer until tender, about 2 minutes. Add the pasta, cooked pancetta, edamame peas, petite peas, Parmesan, the remaining 1/4 cup olive oil, salt, and mint leaves. Toss until coated, adding the reserved pasta water, 1 tablespoon at a time, to loosen the sauce. Transfer to a large bowl and serve.
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