Place each chicken breast between two pieces of plastic wrap or parchment paper. Use a meat mallet or rolling pin to gently pound the chicken to an even thickness of about 1/2 inch.
In a shallow dish, combine the flour, salt, and black pepper. Dredge each chicken breast in the flour mixture, shaking off any excess.
Heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Once hot, add two chicken breasts to the skillet. Cook for about 4-5 minutes on each side until golden brown and cooked through. Remove the chicken from the skillet and set aside on a plate. Repeat with the remaining chicken and oil.
In the same skillet, add the sliced mushrooms and sauté for about 5 minutes until they are browned and tender.
Pour in the Marsala wine, scraping any brown bits from the bottom of the skillet. Allow the wine to simmer for 2-3 minutes until slightly reduced.
Add the chicken broth and bring the mixture to a simmer. Return the cooked chicken to the skillet and cook for an additional 5 minutes, allowing the chicken to absorb the flavors of the sauce.
Stir in 2 tablespoons of butter until melted and combined with the sauce. Remove from heat.
Serve the chicken marsala hot, spooning the sauce and mushrooms over the top. Garnish with chopped fresh parsley.