In a medium bowl, toss the mushrooms with 1/2 teaspoon salt and a few grinds of pepper. Heat the oil and butter in a large skillet over medium-high heat until the butter melts and most of its foam cooks off, about 1 min. Meanwhile, season the chicken with 3/4 teaspoon salt and a few generous grinds of pepper. Spread the flour on a large plate and lightly dredge the chicken. Saute the chicken pieces, flipping after 2 minutes, until they're lightly brown on both sides, about 4 minutes total (it's all right if the sides of the chicken are still raw). Add the mushrooms and cook, stirring, until they start to soften and brown in places, about 3 minutes. Add the chicken broth, caramelized onions, and sun-dried tomatoes. Bring to a boil and then reduce to a gentle simmer. Cover and cook until the chicken is firm to the touch and cooked through (check by slicing into one of the thicker pieces) and the mushrooms are tender, about 6 minutes. Stir in the vinegar, the Tabasco, and half of the parsley. Taste for salt and pepper and serve sprinkled with the remaining parsley.

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.