Preheat the oven to 400°F (204°C). Line a baking sheet with foil or a shallow roasting pan.

In a skillet over medium heat, melt butter with olive oil. Add mushrooms and cook, stirring, until they release liquid and it evaporates (about 6–8 minutes). Add garlic and thyme for the last minute. Remove from heat and let cool slightly.

Stir the Parmesan, half the mozzarella, and parsley into the cooled mushrooms. Season to taste.

Lay each breast flat and slice horizontally almost through, opening like a book. Pound gently if needed to even thickness (about 1/2 inch).

Spoon an even amount of filling into each breast. Fold closed and secure with toothpicks or tie with twine. If using prosciutto, wrap it around the breast now.

Season the outside of the chicken with salt and pepper.

Heat a tablespoon of olive oil in an oven-safe skillet over medium-high heat. Sear breasts 2–3 minutes per side until golden brown.

Transfer the skillet or place breasts on the baking sheet and bake for 12–18 minutes, depending on thickness, until an instant-read thermometer reads 165°F (74°C) in the thickest part.

If you prefer a cheesier top, sprinkle remaining mozzarella over the breasts for the last 3–5 minutes of baking to melt and brown slightly.

Let chicken rest 5 minutes before slicing so juices redistribute.