Bring a large pot of well-salted water to a boil. Meanwhile, put 2 tablespoons of the oil and the pancetta in a 12-inch skillet or saute pan and cook over medium heat, stirring, until the pancetta begins to brown and renders much of its fat, 5 to 7 minutes. Using a slotted spoon, transfer the pancetta to a plate lined with paper towels. Raise the heat to medium high, add the asparagus, and season with 1/4 teaspoon salt. Stir often until the asparagus starts to brown, about 3 minutes. Add the caramelized onions and chicken broth, reduce the heat to medium low, cover, and cook until the asparagus is tender, about 4 minutes. Stir in 1/4 teaspoon pepper and set aside in a warm spot.

Cook the pasta in the boiling water, stirring often, until it's just tender, about 11 minutes. Reserve 1/4 cup of the pasta water. Drain the pasta and add it to the pan with the asparagus. Set the pan over medium-high heat and toss well. Add 1/2 cup of the Parmigiano, the vinegar, and the cooked pancetta and continue cooking, stirring, for 1 minute to meld all the flavors. If the pasta begins to dry, add the reserved pasta water. Serve with a drizzle of the remaining 1 tablespoon olive oil and a sprinkling of the remaining Parmigiano.

Heat the oil in a 12-inch skillet over medium-high heat until it immediately bubbles when an onion touches it, about 1 minute. Add the onions, sprinkle with the salt, and cook, stirring frequently, until they wilt completely and begin to stick to the bottom of the pan, 10 to 20 minutes (much of the onions' moisture will evaporate and the pan will begin to brown; the cooking time varies with the onions' moisture content). Reduce the heat to medium low and cook, stirring and scraping the pan with a wooden spoon every few minutes. (If the pan begins to look like it's burning, add 2 tablespoons water, stir, and lower the heat a bit.) Cook, stirring and scraping and adding water as needed, until the onions are a uniform caramel brown, another 30 to 45 minutes. (If they haven't begun to brown much after 20 minutes, raise the heat to medium.) Add a couple of tablespoons of water and scrape the pan well. Use right away or spread the onions on a baking sheet and let cool to room temperature. Store in the refrigerator for up to one week.

For even slices, use a radial cut: One way to ensure the most even browning is to use a radial cut when slicing the onions. By slicing onion halves on an angle, rather than straight down, you get pieces of similar length and width that will cook at the same rate.