Using a rolling pin or mallet, pound the chicken breast fillets flat until they are about ¼" thick. You want them really thin.
Season each side of the chicken breasts with salt and pepper. Place the ½ cup of flour into a shallow bowl. Then dredge each piece of chicken into the flour, coating all sides.
Heat 3 tablespoons of olive oil in a large skillet over medium-high heat. Add chicken to the skillet, making sure not to crowd the chicken. Work in batches if you need to. Cook for about 3 minutes on each side, or until each chicken is slightly golden brown. Transfer the cooked chicken to a plate and set aside.
Add the remaining 2 tablespoons olive oil and 4 tablespoons butter to the skillet.
Add the onions and the garlic. Stirring slowly, cook over medium heat for about 1 to 2 minutes to soften the onions and garlic.
Add 1 tablespoon of flour, then slowly add the Marsala and chicken stock to the skillet. Use a wooden spoon to stir the onions and garlic in the liquid and also to gently scrap up the bottom of the skillet. Cook the liquid and keep stirring for about 2 minutes, until the flour is fully incorporated with the other ingredients.
Add mushrooms and cook until golden brown, about 5 to 7 minutes, stirring constantly to ensure all of the mushrooms are properly coated in the sauce. The sauce will start to thicken.
Return the chicken to the pan, along with remaining 4 tablespoons of butter. Cook for about 2 minutes, gently tossing the ingredients to coat everything well in the sauce.
Sprinkle in the basil, stir and cook for another minute or two, and then remove from heat.
Sprinkle with parsley and serve immediately.
Notes
For added flavor, top with freshly grated parmesan cheese and serve with a lemon wedge.Make it gluten-free: use almond flour instead of all-purpose flour.Storage: store in an airtight container in the refrigerator for up to 4 days.