Preheat your oven to 400°F.
In a 10-inch skillet, melt the butter over medium heat.
Add the onion and garlic to the skillet and cook until softened, about 3 minutes.
Stir in the mushrooms and cook until their juices are released and reabsorbed, about 5 minutes.
Optional: Add white wine and thyme, cooking until evaporated but not dry.
Transfer the mixture to a bowl and let it cool completely.
Combine cooled mushrooms with cream cheese, mozzarella cheese, and a pinch of salt and pepper.
Using a paring knife, cut a 1 ½-inch slit along the side of each chicken breast to create a pocket.
Divide the mushroom mixture into four portions and fill each pocket, securing with toothpicks if necessary.
Mix garlic powder, paprika, ½ teaspoon salt, and ¼ teaspoon pepper in a small bowl. Season the outside of the chicken.
Heat olive oil in an ovenproof skillet over medium heat. Sear the chicken on both sides until golden, about 3 minutes each side.
Transfer the skillet to the oven and bake for 17-21 minutes, or until the chicken reaches an internal temperature of 165°F.
Remove the chicken to a plate and let it rest for 5 minutes before serving.