In a large mixing bowl, combine 2 T olive oil, 2 T sherry vinegar, the honey, mustard, red-pepper flakes, Kosher salt and pepper Whisk until smooth Add chicken pieces to the mixture, coating it well Set aside at room temperature, stirring it once Heat a 12-inch cast-iron or heavy skillet over medium-high until very hot, 1 1/2 to 2 minutes Add the onions, season with salt, then cook, mostly undisturbed, for 4 minutes more, stirring every minute or so Add the mushrooms and stir to combine 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 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 mix together the chicken and vegetables Cook, stirring occasionally, until the chicken is cooked through, about 10 minutes more Add the remaining 1 T sherry vinegar, stirring and scraping up anything on the bottom of the skillet Season to taste with salt Remove from the heat and top with the parsley and cheese Serve with bread or pasta