Preheat oven to 390F (200C), or 355F for a fan-forced oven. Cut a 3 to 4 inch wide pocket into the side of each chicken breast, going almost to the edges without cutting all the way through. Pat the chicken dry with paper towels. 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 they start turning golden. Add garlic, thyme, and remaining salt and pepper. Cook for another 2 minutes. Add baby spinach and stir until wilted, about 30 seconds. Remove filling from pan and let cool slightly.

Stuff the mushroom mixture evenly into each chicken breast pocket. Lay mozzarella slices on top of the filling inside the pocket. Seal the openings with toothpicks. Mostly closed is enough.

Wipe the skillet clean with paper towels. Heat olive oil over medium-high heat. Sear each stuffed chicken breast for about 1 and a half minutes per side until golden brown. Do not move the chicken while searing.

Transfer the skillet to the preheated oven. Bake for about 15 minutes, or until a meat thermometer reads 149F (65C) in the thickest part of the chicken flesh.

Remove from the oven, place on a plate, and cover loosely with foil. Rest for 5 minutes before serving.