Preheat the oven to 350 degrees F.

In a medium bowl, combine all the stuffing ingredients and mix until combined.

Take each chicken breast, pat it dry and season it lightly on both sides with salt and pepper. Butterfly the chicken through the thickest part (don't slice all the way through) so it opens up into one large, thinner piece of chicken (see pictures below, if needed).

Divide the filling into six portions. Press the stuffing onto one side of each chicken breast, covering as much of that side as possible and compressing the stuffing lightly with your hands so it stays together. Fold the other side of the chicken over tightly. It doesn't need to seal completely.

In a large nonstick skillet, heat the butter and olive oil over medium to medium-high heat until it is hot and rippling. Add the chicken, top-side down to the hot oil in a single layer without crowding the pan too much (you'll probably need to do this in two batches) and brown on each side for about a minute, flipping carefully. Repeat with the remaining chicken.

Place the browned chicken in a lightly greased 9X13-inch baking dish. Bake for 20-30 minutes until the chicken is baked through and the juices run clear (the center of the chicken, not the cheese stuffing, should reach about 160 degrees F on a thermometer).

While the chicken bakes, add the mushrooms to the hot skillet (don't clean it out beforehand, let the mushrooms cook in the remaining oil/butter), scraping up any bits of chicken on the bottom of the skillet. Sprinkle a little salt and pepper over the mushrooms. Let them cook down, stirring occasionally, until they have released their juices, 5-6 minutes. Crank the heat up a bit to medium-high.

Stir together the balsamic vinegar, grape juice and beef broth. Pour into the skillet and use a wooden spoon to scrape up any bits on the bottom of the pan. Simmer the sauce until it reduces by half. This may take a while, 20 minutes or so, but be patient - it will concentrate the flavor and help thicken. Stir in the cream. Season with salt and pepper to taste.

Serve immediately; place a chicken breast over a large scoop of garlic, mashed potatoes and top with a spoonful or three of the mushroom sauce.