Slice each chicken breast in half horizontally to make 4 thinner cutlets, then place between sheets of plastic wrap and gently pound to an even thickness.
Season both sides of the chicken with 1 teaspoon seasoned salt and 1/2 teaspoon ground black pepper.
Heat 1 tablespoon butter in a large skillet over medium-low heat. Add the sliced onion and a two-finger pinch of salt, and sauté for 3–4 minutes until starting to soften.
Add the sliced mushrooms to the skillet and continue to sauté 3–4 minutes until the mushrooms begin to brown; remove onions and mushrooms to a plate.
If the skillet is dry, add 1 teaspoon butter, then increase heat to medium. Add chicken (work in batches if needed) and cook 6–8 minutes on the first side without moving.
Flip the chicken and cook 3–4 minutes more until cooked through and juices run clear (internal temperature 165°F/74°C).
Top each cooked chicken piece with the sautéed onions and mushrooms, then sprinkle with shredded Monterey Jack cheese. Turn off the heat and cover the skillet briefly until the cheese melts, about 1–2 minutes.
Serve immediately.
