In a large mixing bowl, combine 2 T olive oil, 2 T sherry vinegar, 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 once
Heat a large, 12-inch heavy skillet over medium-high Add onions, season with salt, then cook, mostly undisturbed, for 4 minutes more, stirring every minute or so Add mushrooms and stir to combine Allow to cook mostly undisturbed until mushrooms shrink and start to brown, about 4 minutes, stirring every minute or so
Stir in remaining 2 T olive oil Allow onions to cook until they start to color, stirring and lowering the heat as necessary to avoid burning, about 2 minutes
Push onions and mushrooms to edges of skillet, then add chicken pieces to the center Pour any remaining marinade (there will be very little) over onions and mushrooms Cook undisturbed for 4 to 5 minutes, then mix together the chicken and vegetables Cook, stirring occasionally, until chicken is cooked through, about 10 minutes more
Add remaining 1 T sherry vinegar, stirring and scraping up anything on the bottom of the skillet Season to taste with salt Remove from heat and top with parsley and cheese Serve with bread or pasta -
