Add the flour to a plate and coat the chicken breasts in flour. In a large saucepan, heat olive oil to medium-high heat. Fry chicken on both sides until golden brown, about 3 minutes per side. The goal is to get a crispy brown crust. Set aside on a cooling rack.
Lower the heat to medium. Add mushrooms to the pan. DON'T stir the mushrooms. Let them sit and brown for 3-5 minutes. Then stir and allow them to brown again for another 3-5 minutes. Then add the shallots and saute until soft.
Pour in the marsala wine and chicken broth and simmer for 7 minutes until the sauce begins to thicken. Add the chicken breast into the sauce and simmer until the chicken reaches 165℉.
Remove the chicken from the pan and set aside. Stir in the lemon juice and taste. Then add a pat of butter and mix until combined.
Pour your sauce over the chicken and serve.
Notes
Pounding the chicken breasts is important so that they fry evenly and don't burn on the edges while trying to get them to 165 degrees. Use a meat tenderizer if you have one but if not, cover them with saran wrap, place a tray on top of them, and pound them with your hand.
Quality marsala wine is important here. It will define the taste of your dish. Look for it in the wine section and don't buy "cooking marsala." The rule of thumb is if you wouldn't drink it, don't put it in your food.
Browning the mushrooms. Really let the mushrooms sit for a while. The more you stir them the more steam you will create. Then you'll have rubbery mushrooms, and no one likes that! You also don't need a lot of oil for mushrooms because they have so much liquid inside them. After you fry the chicken, you shouldn't need to add any more oil.