Preheat oven to 350 degrees. Lightly spray a 13 x 9-inch baking dish.
Mix the flour, 1 tsp salt and 1/4 tsp of pepper on a plate. Set aside.
Butterfly chicken breasts. Here's a good video if you don't know how:
Pound each butterflied breast between 2 sheets of plastic wrap using a meat mallet to 1/4-1/2-inch thickness. Set aside.
Mix all filling ingredients together in a large bowl; stir well to combine. Divide the filling into 4 equal balls.
Lay chicken breast on a plate. Place a ball of the cheese filling on one "wing" of the chicken breast (butterflied chicken breast will have 2 "wings" like a butterfly). Flatten the cheese filling into a shape like a hockey puck. Fold the other "wing" over the filling--it's not necessary to seal. Repeat for other chicken breasts.
Heat a large skillet over medium-high heat. Add the olive oil and heat until the oil shimmers.
Dredge each chicken breast in the season flour; place chicken breast in hot skillet and brown well on each side--you may have to brown in shifts. Don't wash the pan--you will use this to make the sauce and you want the chicken "drippings" for flavor.
Place chicken in prepared baking dish.
Bake for 20-30 minutes (or until internal temp is 165 degrees F or juices run clear). While the chicken bakes, prepare the sauce.
To make sauce, heat the same skillet you browned the chicken in over medium heat. Add the sliced onion and saute for 5 minutes. Add the mushrooms and saute another 5 minutes.
Deglaze the pan with the wine, scraping up all the browned bits from the chicken. Add the cream and simmer over medium-low heat until sauce is reduced by 50% (this should take 5-10 minutes). Add salt and pepper to taste. Spoon over chicken.