Preheat oven to 390 degrees F (200 degrees C), or 355 degrees F for a fan-forced oven. Using a sharp knife, cut a deep pocket into the thickest side of each chicken breast, 3 to 4 inches wide, going nearly to the edges without cutting through. Season inside and outside with salt and pepper.

Melt butter in an oven-proof skillet over high heat. Add mushrooms and cook for about 3 minutes until golden. Add garlic, thyme, and a pinch of the remaining salt and pepper. Cook another 2 minutes. Add baby spinach and stir until just wilted, about 30 seconds. Remove filling from pan and set aside to cool slightly.

Spoon the mushroom mixture into each chicken breast pocket. Layer mozzarella slices on top of the filling. Secure openings with 2 to 3 toothpicks per breast. They do not need to be perfectly sealed.

Wipe the skillet clean with paper towels. Heat olive oil over medium-high heat until shimmering. Sear each stuffed breast for about 1 and a half minutes per side until deeply golden brown.

Transfer skillet to the preheated oven. Bake for about 15 minutes, or until a meat thermometer reads 149 degrees F (65 degrees C) in the thickest part of the chicken flesh. Do not probe the filling.

Remove from oven, transfer to a plate, and tent loosely with foil. Rest 5 minutes before serving. Serve warm with your choice of sides.