Prepare chicken by pounding to an even thickness
In a shallow bowl or a pie plate, combine the 1/2 cup of all-purpose flour with the 1 teaspoon of salt and 1/2 teaspoon of black pepper.
Pat the pounded chicken dry with paper towels.
Dredge each piece of chicken in the flour mixture, shaking off any excess. You want a light, translucent coating, not a thick batter.
Place a large stainless steel or cast-iron skillet over medium-high heat.
Add the 2 tablespoons of vegetable oil and the first 2 tablespoons of unsalted butter.
Once the butter is foaming and the pan is hot, add the chicken in a single layer. Do not overcrowd the pan
Sear the chicken for 3 to 4 minutes per side until deeply golden brown. Use a calibrated food thermometer to check that the internal temperature has reached 165°F.
Remove the chicken to a plate and tent loosely with foil to keep warm.
In the same skillet (do not wipe it out!), add the remaining 2 tablespoons of butter.
Once melted, add the sliced mushrooms. The mushrooms will initially absorb the fat and then release their moisture.
Resist the urge to move them constantly. Let them sit for 2 minutes to develop color, then stir.
Once the mushrooms are browned and their liquid has evaporated, add the minced garlic.
Sauté for only 30 to 60 seconds until fragrant, being careful not to burn the garlic
Pour the 3/4 cup of dry Marsala wine into the pan. Use a wooden spoon or a whisk to scrape the bottom of the skillet vigorously.
Add the 1/2 cup of chicken broth. Increase the heat slightly to bring the liquid to a simmer.
Allow the sauce to reduce by about half, which should take 5 to 7 minutes.
Once the sauce has thickened and the sharp smell of raw alcohol has dissipated, turn the heat to low.
Return the chicken and any accumulated juices on the plate back into the skillet.
Spoon the sauce over the chicken and let it simmer for 1 to 2 minutes to reheat the protein and marry the flavors.
Taste the sauce. If it needs more salt or a hit of acid, add a tiny pinch of salt or a drop of lemon juice (optional).
Garnish with the fresh chopped parsley.
Serve immediately over a bed of egg noodles, mashed potatoes, or alongside crusty bread to soak up every drop of the sauce.