Bring a large pot of well-salted water to a boil. Meanwhile, put 2 Tbs. oil and the prosciutto in a large (12-inch) skillet, place over medium heat, and cook, stirring occasionally, until the prosciutto browns in places and becomes crisp, about 5 minutes. Transfer the prosciutto to a large plate lined with paper towels.
Add 1 Tbs. oil and the onion to the skillet, sprinkle with 1/2 tsp. salt, and cook, stirring occasionally, until the onion softens completely and turns light brown, about 6 minutes; add 1 or 2 Tbs. water to the skillet if the onion starts to stick or burn. Add the zucchini and corn, sprinkle with 1/4 tsp. salt, and cook, tossing occasionally, until the zucchini becomes tender, 4 to 5 minutes. Remove from the heat and stir in half the Pecorino-Romano and all the mint.
Add the penne to the pot of boiling water and cook according to the package directions. Reserve 1/2 cup of the pasta water and then drain the pasta. Add the pasta, the remaining 2 Tbs. oil, the vinegar, and 1 tsp. black pepper to the skillet with the zucchini and corn mixture. Set the skillet over medium heat and cook, stirring, for 1 minute so the pasta mixes with the vegetables. Add the reserved pasta water and stir. Serve sprinkled with the crisp prosciutto and the remaining Pecorino-Romano.
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