Heat 1 tablespoon olive oil in a medium sauté pan over medium-high heat. Add minced shallot and cook for 2 minutes until translucent. Add garlic and cook for 30 seconds until fragrant.

Add chopped mushrooms, ¼ teaspoon salt, and ¼ teaspoon pepper. Cook, stirring occasionally, until mushrooms release their moisture and it evaporates, approximately 8-10 minutes. Mushrooms should be well-browned and reduced in volume by about 65%.

Add white wine if using and scrape up any browned bits. Cook until liquid evaporates, about 1-2 minutes. Transfer to a bowl and let cool for 10 minutes.

Once mushroom mixture has cooled slightly, stir in 4 oz (1 cup) shredded Swiss cheese, fresh thyme, and Dijon mustard. Mix thoroughly to combine.

Place chicken breasts on a cutting board. With your palm flat on top of each breast, carefully slice horizontally into the thicker side, creating a pocket while leaving the opposite side intact (like opening a book). Do not cut all the way through.

Open each chicken breast and season the interior with a pinch of salt and pepper. Divide the mushroom-cheese mixture evenly among the chicken breasts. Fold the top flap over to close. Secure with toothpicks or kitchen twine. Season the exterior with remaining salt and pepper.

Preheat oven to 375°F (190°C).

Heat a large oven-safe skillet over medium-high heat. Add 2 tablespoons butter. Once butter is foaming, add the stuffed chicken breasts and sear until golden brown, about 3-4 minutes per side.

Transfer the skillet to the preheated oven and bake until chicken reaches an internal temperature of 165°F (74°C), approximately 15-18 minutes depending on thickness.

Remove from oven and transfer chicken to a cutting board. Let rest for 5 minutes to allow juices to redistribute. Remove toothpicks or twine before slicing and serving.