In a spacious mixing bowl, blend 2 tablespoons of oil, 2 tablespoons of vinegar, honey, mustard, red pepper flakes, and 1 teaspoon of salt. Whisk the mixture until it forms a smooth consistency. Pat the chicken pieces dry and season them with salt and pepper. Then, add the seasoned chicken to the mix, ensuring it is thoroughly coated. Leave it at room temperature, stirring it once, while you proceed to prepare the onions. Begin by preheating a 12-inch cast-iron or sturdy skillet over medium-high heat until it becomes very hot, which should take approximately 1½ to 2 minutes. Once hot, add the thinly sliced onions in an even layer, season them with salt, and cook for 4 minutes. During this time, gently stir the onions every minute or so. Afterwards, add the sliced mushrooms, season them with salt, and stir to combine. Don't worry if it looks crowded; this is perfectly fine. Allow the mixture to continue cooking, with occasional stirring, until the mushrooms shrink and acquire a golden brown hue, which usually takes about 4 minutes.
Now, add the remaining 3 tablespoons of olive oil to the skillet and cook the onions, stirring occasionally, until they develop a richer color. Be mindful of lowering the heat to prevent burning, which should take approximately 2 minutes.
Next, gently push the onions and mushrooms toward the edges of the skillet, creating space in the center. Place the chicken pieces in the center of the skillet. Pour any remaining marinade (which will be minimal) over the onions and mushrooms.
Allow the chicken to cook undisturbed for 4 to 5 minutes, then combine it with the vegetables. Continue occasionally stirring until the chicken is thoroughly cooked, typically around 10 minutes more. If the onions start to brown too quickly, reduce the heat to medium to prevent burning.
Add the remaining 2 teaspoons of sherry vinegar, stirring and scraping up any bits from the bottom of the skillet. Season the mixture to taste with salt. Take the skillet off the heat and garnish it with the chopped parsley and, if desired, grated cheese. Serve this delightful dish with either bread or pasta.