Pat chicken breasts dry, and if necessary, cut in half horizontally.* Place each breast or cutlet on a cutting board and pound with a mallet until about ¼ inch thick.
In a shallow dish, combine flour, ½ teaspoon salt, and ½ teaspoon pepper. Lightly dredge chicken in flour mixture. Set aside. Alternatively, add the flour, salt, and pepper to a zip top bag, add chicken, and shake to coat.
To a large skillet on medium high heat, heat oil until shimmering. Cooking in batches, add the chicken in one layer, adding oil as needed. Brown for 2-3 minutes on each side. Remove chicken from the pan and set aside.
To the same skillet on medium heat, add butter, shallot, and garlic if using, and cook for about 3 minutes or until shallot is almost translucent. Add mushrooms and cook for 5 minutes or until juices are released.
Add wine and bring to a boil. Reduce heat and simmer for 3-5 minutes, or until slightly thickened.
Add chicken broth, heavy cream, ½ teaspoon salt, and ¼ teaspoon pepper. Heat until bubbling, and simmer 3-5 minutes until slightly thickened.
Add chicken into the sauce and simmer for 2-3 minutes or until chicken is heated through. Garnish with parsley and serve immediately over pasta, mashed potatoes, or rice.
Notes
*For very thick breasts, pat dry and place flat on a cutting board. With knife parallel to the board, cut each breast in half horizontally through the middle to make 2 slices, each slice being half as thick. Use a paper towel and the flat of your hand to hold the breast in place as you cut. Place each slice between 2 layers of plastic wrap. Pound with a mallet until ¼ inch thick.
Leftovers should be refrigerated in an airtight container for up to 3 days, or frozen for up to 4 months.