Preheat oven to 400°F (200°C).

In a medium skillet over medium-high heat, melt butter with 1 tablespoon of olive oil. Add the chopped mushrooms and cook until browned, about 5-7 minutes. Stir in the minced garlic and fresh thyme and cook for 1 minute more until fragrant. Remove from heat and let cool slightly.

In a mixing bowl, combine the cooked mushroom mixture with the shredded smoked Gouda. Season with a pinch of salt and pepper.

Place each chicken breast on a cutting board and carefully slice horizontally through the thickest part to create a pocket, without cutting all the way through. Open the breast like a book. Gently pound to an even thickness (about 1/4 inch).

Season both sides of the chicken with salt, pepper, and paprika. Spoon the Gouda and mushroom mixture evenly into the pocket of each chicken breast. Fold the chicken over and secure the edges with toothpicks.

Heat the remaining olive oil in a large, oven-safe skillet over medium-high heat. Sear the stuffed chicken breasts for 3-4 minutes per side, until golden brown.

Transfer the skillet to the preheated oven and bake for 15-20 minutes, or until the chicken is cooked through and the internal temperature reaches 165°F (74°C).

Remove from the oven, remove the toothpicks, and let the chicken rest for 5-10 minutes before slicing and serving.