Preheat oven to 375°F (191°C) with rack in lower-middle position. Pat chicken breasts completely dry with paper towels.
In a bowl, combine mozzarella, provolone, Parmesan, sun-dried tomatoes, sour cream, garlic powder, Italian seasoning, and salt and pepper. Mix until cohesive.
Cut a horizontal pocket into each chicken breast, keeping 1/2 inch intact on edges. Season inside pockets with salt and pepper.
Stuff each pocket with 2-3 tablespoons of cheese mixture. Press edges to seal and secure with 2-3 toothpicks along the open edge.
Dredge each stuffed breast in flour seasoned with salt and pepper, shaking off excess.
Heat olive oil in oven-safe skillet over medium-high heat. Sear chicken 3-4 minutes per side until golden brown. Transfer to a plate.
In same skillet, melt 2 tablespoons butter over medium heat. Sauté onions 3-4 minutes, add garlic for 1 minute, then cook mushrooms until browned and tender.
Pour in Marsala wine, scraping up browned bits, and simmer 2-3 minutes. Add chicken broth and simmer 2-3 minutes. Stir in heavy cream and thyme; season with salt and pepper.
Return chicken to skillet, spoon some sauce over top, and transfer to preheated oven. Bake 18-22 minutes until chicken reaches 165°F (74°C) internally.
Remove toothpicks, let chicken rest 3-5 minutes. Spoon sauce over chicken, garnish with parsley, and serve immediately.