Preheat oven to 375°F (190°C).
In a skillet, melt butter with olive oil over medium heat. Add onions and cook until caramelized, about 15–20 minutes, stirring occasionally. Add garlic, thyme, salt, and pepper in the last 2 minutes of cooking.
Place chicken breasts in a lightly greased baking dish. Pour caramelized onions over the chicken.
Mix broth with sour cream (if using) and pour over the chicken and onions.
Sprinkle shredded cheese evenly on top. Add breadcrumbs if desired for a crispy topping.
Bake uncovered for 25–30 minutes, or until chicken is cooked through and cheese is golden and bubbly.
Garnish with fresh parsley and serve hot.
Serving and Storage Tips:
