Place chicken breasts on a cutting board. (Note: If your chicken breasts are thick, you may only need two for this recipe. You can slice them in half horizontally.) Cover the chicken with plastic wrap. Pound with a meat mallet until they're about a 1/4 inch thick.
Season the chicken all over with salt and pepper. Coat each chicken breast with the flour, shake off any excess, and set them on a plate.
Heat 1 tablespoon of olive oil and 2 tablespoons of butter in a large skillet on medium-high heat. When hot enough for the chicken to sizzle, place the chicken breasts in the pan. Cook for about 3 minutes on each side until golden brown and internal temperature reaches 155° F (they will cook a little more in the sauce). Remove from pan and set aside.
Melt a tablespoon of butter on medium heat in the pan used to cook the chicken. If the pan feels dry, add more olive oil. Add mushrooms and cook for 3 minutes or until beginning to brown.
Add shallots and salt to taste. Cook until shallots soften (about 2 minutes). Add the garlic and stir frequently until garlic becomes fragrant (about 1 minute).
Add chicken broth, heavy cream, Marsala wine, and thyme. Season with salt and pepper. Bring to a boil, reduce heat, and allow to simmer for 10 minutes.
Once sauce has reduced, add the chicken to the pan. Simmer in the sauce on low for another 5 minutes until sauce has thickened a bit more.
Garnish with parsley.