If your chicken breasts are thick, pound them to 1/2-inch even cutlets — this is the 30-minute enabler: thin cutlets cook through in about 6 minutes total. Season both sides with half the salt and pepper, then lightly dredge in the flour, shaking off the excess.

Heat 1 tablespoon of the oil and 1 tablespoon of the butter in a large skillet over medium-high heat until the butter foams. Sear the cutlets in a single layer, about 3 minutes per side, until golden and cooked to 165°F internally at the thickest point. Work in two batches if needed — crowded chicken steams instead of searing. Remove to a plate.

Add the remaining 1 tablespoon of oil to the same skillet. Add the mushrooms in a single layer and let them sit undisturbed for 2–3 minutes so they brown hard, then stir and cook 2 more minutes until deeply golden.

Stir in the shallot, garlic, thyme, and the remaining salt and pepper; cook 2 minutes until fragrant. Pour in the Marsala wine, scraping up the browned bits from the pan, and let it bubble and reduce by about half, about 2 minutes.

Pour in the chicken broth and bring to a simmer. Reduce the heat to medium-low and stir in the half-and-half. Simmer gently, stirring occasionally, until the sauce thickens enough to coat the back of a spoon, about 4–5 minutes. Do not boil hard — a gentle simmer keeps the half-and-half silky instead of splitting.

Return the chicken (and any juices on the plate) to the skillet and spoon the sauce over the top. Simmer 1–2 minutes until the chicken is heated through. Swirl in the remaining 1 tablespoon of butter, shower with parsley and extra thyme, and serve over mashed potatoes or pasta with extra sauce spooned over everything.