Preheat oven to 375 F. Season the chicken breasts. Mix flour, oregano, Kosher salt, and pepper in a small bowl, then whisk to combine. Place flour mixture in a large Ziploc bag, then add chicken breasts. Seal and shake to thoroughly coat on all sides.
Cut a pocket into chicken breasts. Use a sharp knife to cut 2-3" wide pocket into the thick side of each chicken breast, being sure not to cut all the way through the other side.
Brown the chicken. Heat a large skillet over medium-high heat, then add 2 Tbsp butter and 2 Tbsp olive oil. Add seasoned chicken breasts and cook for 7-8 minutes, or until golden brown. Carefully flip, then cook for another 3-5 minutes on the other side. Transfer chicken to a large plate.
Sauté onions and mushrooms. Add another 2 Tbsp of butter or olive oil, then add sliced onions. Cook for 3-5 minutes, stirring often, until onions are starting to soften and brown slightly. Add mushrooms to the pan and cook an additional 5 minutes until browned.
Reserve filling for stuffing. Remove 1 heaping cup of sautéed onions and mushrooms, then transfer to a small mixing bowl. Allow to cool slightly.
Make the marsala sauce. To the skillet, add ¾ cup marsala wine, ¼ cup sherry vinegar, and 2 Tbsp balsamic vinegar, then scrape the bottom of the pan with a spatula or wooden spoon. Bring to a boil, then lower heat to a simmer. Allow sauce to reduce while you stuff the chicken.
Make the stuffing. To the cooled onion and mushrooms, add grated fontina cheese. Stir to mix well, then divide and stuff evenly between chicken breasts. It's okay if it overflows a bit. Really fill 'em!
Bake the stuffed chicken marsala. Place stuffed chicken breasts back in the skillet, spoon some of the sauce on top, then transfer to a preheated oven. Bake for 12-14 minutes, or until chicken has reached an internal temperature of 165 F (74 C).
Serve immediately. Sprinkle with grated Parmesan cheese, then sprinkle with fresh chopped parsley. Serve immediately!