Preheat the oven to 200°C/390°F or 180°C/350°F fan.
Pat the chicken breasts dry with paper towels. Using a sharp knife, carefully cut a pocket into the thickest side of each breast. Avoid cutting completely through the chicken.
Season the inside and outside of the chicken with half of the salt and pepper.
Place a large oven-safe skillet over medium-high to high heat. Add the butter and let it melt.
Add the sliced mushrooms and cook for about 3 minutes, stirring occasionally, until they begin to brown.
Add the garlic, thyme, remaining salt, and pepper. Cook for another 2 minutes until the mushrooms are deeply golden and fragrant.
Add the baby spinach and stir for about 30 seconds, just until wilted. Remove the mushroom mixture from the heat.
Spoon the mushroom and spinach mixture into each chicken pocket. Add the mozzarella slices on top of the filling.
Secure the openings loosely with toothpicks. The chicken does not need to be completely sealed.
Carefully wipe out any excess filling or moisture from the skillet. Add the olive oil and heat over medium-high heat.
Add the stuffed chicken and sear each side for about 1 1/2 minutes, or until nicely golden. This gives the mushroom stuffed chicken breast a flavorful browned exterior.
Transfer the skillet to the preheated oven and bake for approximately 15–20 minutes, depending on the thickness of the chicken.
Check the thickest part of the chicken with an instant-read thermometer. The chicken should reach 74°C/165°F for safe consumption.
Remove the chicken from the skillet and let it rest loosely covered with foil for about 5 minutes.
Carefully remove the toothpicks before serving.