Pound the chicken to an even 3/4-inch thickness and season it with the salt and pepper.
Heat the oil in a large skillet over medium-high heat and brown the chicken for 4 minutes per side, then move it to a plate.
Add the butter, mushrooms and onion, and cook for 8 to 10 minutes until the liquid evaporates and the vegetables develop brown edges.
Pour in the broth and Worcestershire sauce, scraping the skillet, and simmer for 2 minutes.
Return the chicken and any juices to the pan, cover and cook over medium-low heat for 5 to 7 minutes until the thickest center reaches 165°F.
Divide the cheese over the chicken, cover for 1 minute until melted, and spoon the mushrooms, onions and sauce around each piece.