Slice each chicken breast in half lengthwise to create 4 thinner portions.
Pierce the chicken all over with a fork, then place it in a resealable bag with the Italian dressing.
Refrigerate for at least 4 hours and up to 24 hours.
Preheat the oven to 400°F. Slice the mushrooms and onion before you begin cooking.
Heat a small amount of oil in an oven-safe skillet over medium-high heat.
Remove the chicken from the marinade and sear it on both sides until lightly charred and browned. Watch closely because the Italian dressing can cause the chicken to color quickly. Transfer the chicken to a plate before it is cooked through.
Reduce the heat to medium and add the sliced mushrooms to the same skillet.
Cook until the mushrooms soften and release moisture, using that liquid to loosen the browned bits from the pan. If the pan starts to dry out before enough moisture is released, add up to ¼ cup water.
Add the sliced onion and cook until softened.
Stir in the white pepper, black pepper, garlic powder, and onion powder. Hold the salt until the end.
Once the excess moisture has cooked away, add the beef broth and butter. Stir until the butter has melted.
For a thicker sauce, stir 2 teaspoons flour into the mushrooms and onions before adding the beef broth and butter.
Return the chicken to the skillet and top with the Monterey Jack cheese.
Transfer the skillet to the oven and bake until the cheese is melted and the chicken reaches 165°F in the thickest part. The exact time will depend on the thickness of the chicken.
Taste the mushrooms and onions and add salt only if needed.
Serve the chicken topped with the mushrooms, onions, melted cheese, and pan sauce.