Prep the cutlets: Lay each chicken breast flat and slice it in half horizontally to make 4 thin cutlets. Cover them with plastic wrap and pound to an even 1/4 inch. Pat dry and season both sides with the salt and pepper.

Dredge in flour: Spread the flour on a plate and press each cutlet into it on both sides, shaking off the excess. A thin coat is what browns and what thickens the sauce later. A thick coat turns gummy.

Sear the chicken: Heat the olive oil and 1 tablespoon of the butter in a large skillet over medium-high heat. Cook the cutlets 2-3 minutes per side, in two batches if the pan is crowded, until golden and cooked through to an internal temperature of 165°F. Move them to a plate.

Brown the mushrooms: Add the remaining 2 tablespoons of butter to the same skillet. Add the mushrooms and cook 5-6 minutes without stirring much, until their liquid cooks off and the edges go deep brown.

Add the shallot and garlic: Stir in the shallot and cook 2 minutes until soft, then add the garlic and cook 30 seconds more, just until it smells sweet instead of raw.

Deglaze and reduce: Pour in the marsala and scrape the browned bits off the bottom of the pan. Let it bubble hard for 3 minutes, then add the chicken broth, heavy cream, and thyme. Simmer 10-12 minutes, until the sauce has reduced by about half and coats the back of a spoon.

Finish and serve: Slide the cutlets and any juices from the plate back into the pan, spoon sauce over them, and simmer 2 minutes to warm through. Scatter the parsley over the top and serve.