½ cup fresh flat-leaf parsley or dill leaves and fine stems, roughly chopped
¼ cup grated Parmesan or pecorino (optional)
Bread or cooked pasta, for serving
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In a large mixing bowl, combine 2 tablespoons oil, 2 tablespoons vinegar, the honey, mustard, red-pepper flakes and 1 teaspoon salt; whisk until smooth. Pat the chicken dry and season with salt and pepper, then add to the mixture, coating it well. Set aside at room temperature, stirring it once while you make the onions.
Heat a 12-inch cast-iron or heavy skillet over medium-high until very hot, 1 1/2 to 2 minutes, then add the onions in an even layer. Season with salt, then cook, mostly undisturbed, for 4 minutes more, stirring every minute or so. Add the mushrooms, season with salt, and stir to combine. (It will look crowded, and that’s OK.) Allow to cook mostly undisturbed until the mushrooms shrink and start to brown, about 4 minutes, stirring every minute or so.
Stir in the remaining 3 tablespoons olive oil and allow the onions to cook until they start to color, stirring and lowering the heat as necessary to avoid burning, about 2 minutes. Push the onions and mushrooms to the edges of the skillet, then add the chicken pieces to the center. Pour any remaining marinade (there will be very little) over the onions and mushrooms. Cook undisturbed for 4 to 5 minutes, then combine the chicken and vegetables and cook, stirring occasionally, until the chicken is cooked through, about 10 minutes more. (Reduce the heat to medium if the onions look like they are burning at any point.)
Add the remaining 2 teaspoons sherry vinegar, stirring and scraping up anything on the bottom of the skillet. Season to taste with salt