Cutlets should be 1/4 inch thick, so you might have to pound them with a mallet if they're too thick. Combine the flour, salt + pepper to taste in a large ziploc bag. Toss the cutlets in and shake the bag to coat them in the flour.
Heat the oil and 2 tbs butter in a large deep skillet. When melted, add the chicken cutlets to the pan and cook on medium for 3 minutes per side - chicken should be golden brown. Remove from pan and set aside.
Melt the remaining 1 tbs butter in the pan and add the mushrooms. Saute them for 2-3 minutes until they soften. Next add the shallot, garlic and season with salt to taste. Cook for a minute or two.
Add the broth, Marsala wine, heavy cream and thyme and bring to a boil. Lower the heat to a simmer and cook your sauce for 10 to 15 minutes until it thickens and browns in color, stirring occasionally.
Add the chicken back to the pan and begin to cover them with the sauce. Heat for a few more minutes and then serve with some fresh parsley.
Notes
This dish pairs perfectly with some pasta or mashed potatoes.