Preheat your oven to 375°F (190°C). Lightly grease a baking dish and set it aside.
Prepare the chicken breasts by making a horizontal pocket. Place each breast on a cutting board, slice about 2/3 of the way through, and open it like a book to form a pocket. If needed, gently pound to an even thickness for even stuffing.
Season the chicken pockets with salt, pepper, paprika, and thyme. This keeps the exterior flavorful and helps the filling shine.
Heat the olive oil in a skillet over medium heat. Sauté the garlic for about 30 seconds until fragrant, then add the mushrooms. Cook until they release their moisture and start to brown, about 4–5 minutes.
Add the spinach to the skillet and cook until it wilts, about 1–2 minutes. If the mixture looks dry, splash in a little wine or stock to deglaze the pan. Remove from heat.
Stir in the cream cheese (or sour cream) into the mushroom-spinach mixture until creamy and well combined. Taste and adjust salt and pepper as needed.
Place a slice of Swiss cheese into each chicken pocket, then spoon generous amounts of the mushroom-spinach filling on top of the cheese, finishing with another slice of Swiss to reinforce the filling.
Secure with toothpicks if needed and place the stuffed chicken in the prepared baking dish. Lightly brush the outside with a little olive oil for a golden bake.
Bake for 25–30 minutes, or until the chicken reaches an internal temperature of 165°F (74°C) and the cheese is melted and bubbling. If you like a crisper exterior, switch to broil for the last 2–3 minutes, watching closely.
Remove from oven and let rest for 5 minutes before serving. This helps the juices redistribute and makes slicing easier.