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Preheat oven to 350°F.
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Stuffing: Combine all the cheese stuffing ingredients in a bowl.
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Chicken: Butterfly the thickest section to create two lobes. Place the pounded chicken breasts on a plate and place the desired amount of stuffing on one half of the chicken breast. Gently press the stuffing down and fold over the other side of chicken breast.
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Preheat a large sauté pan on the stovetop. Add 4 oz. oil; heat oil until shimmering.
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Place approximately 2 cups of flour in a pan and season it to taste with salt and pepper. Dredge the chicken in the flour and shake off excess flour. Place the stuffed chicken breasts in the sauté pan with the preheated oil. Cook each side until golden. When chicken is seared on both sides remove them from sauté pan and place them in a baking pan and into the oven for 10-20 minutes.
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Bake until the juices run clear and the stuffing and center reach a temperature of at least 165°.
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Sauce: Add the onions to the sauté pan. Stir with a spatula. After 2 minutes add the mushrooms; sauté mixture until onions are translucent. Deglaze the sauté pan with the marsala wine (make sure to incorporate particles from the bottom of the pan). Bring the wine to a simmer and add the heavy cream. Simmer the sauce on low heat until reduced by half.
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Place cooked chicken breast on a plate and top with onions, mushrooms, and sauce.