Pound the chicken breasts to an even 1/2-inch thickness.
In a small bowl, mix the salt, pepper, garlic powder, and paprika. Rub this seasoning mix evenly over both sides of the chicken breasts.
Heat the olive oil in a large skillet over medium-high heat, or preheat your grill to medium-high.
Cook the chicken for 5 to 7 minutes per side, until the internal temperature reaches 165 degrees Fahrenheit. Remove the chicken from the skillet or grill and set aside on a plate, covering loosely with foil to keep warm.
Return the cooked chicken breasts to the skillet with the vegetables.
Place one slice of Monterey Jack cheese on top of each chicken breast. Cover the skillet and cook for 1 to 2 minutes, or until the cheese is fully melted and gooey.
Serve the smothered chicken immediately.
Notes
For a different flavor, substitute Swiss cheese or Pepper Jack cheese for the Monterey Jack.
If you prefer a stronger onion flavor, use sweet Vidalia onions.