Preheat the oven to 350°F (175°C).
In a skillet over medium heat, melt the butter. Add the sliced onion and cook, stirring occasionally, until the onion is soft and translucent, about 5 minutes.
Remove the softened onions to a plate and set aside.
Lightly coat the skillet with nonstick spray. Season each chicken breast evenly with paprika, garlic powder, and seasoned salt.
Place the seasoned chicken in the hot skillet and cook over medium-high heat about 5–7 minutes per side, until both sides are golden brown.
Return the cooked onions to the skillet with the chicken.
Pour in the French onion soup and the beef broth. Bring to a gentle simmer and cook 2–3 minutes to let the sauce come together and lightly reduce.
If using a cast-iron skillet, sprinkle the mozzarella over each chicken breast now. If not, transfer the chicken, onions, and sauce to an oven-safe baking dish and top with the cheese.
Bake in the preheated oven until the cheese is melted and the sauce is bubbling, about 10–15 minutes. Serve hot.