Trim any fat from the chicken breasts and slice each horizontally to make two thinner cutlets (or butterfly then flatten). Place between two sheets of plastic wrap and gently pound with a meat mallet or rolling pin until about 1/4 inch thick and even.

Season both sides of each cutlet with 1 tsp salt and 1 tsp black pepper, dividing evenly.

Set up dredging station: put 1 cup flour in a shallow dish, beat 2 eggs in a second dish. Dredge each cutlet in flour, shaking off excess; dip in beaten egg, letting excess drip; then lightly dredge again in flour for a thin, even coating.

Heat a large skillet over medium-high heat. Add 2 tbs olive oil and 1 tbs butter. When butter foams and oil shimmers, add chicken cutlets in a single layer (work in batches if needed). Cook 3 to 4 minutes per side until golden brown and cooked through (internal temp 165°F / 74°C). Transfer cooked cutlets to a warm plate and tent with foil.

Sauce

In the same skillet, reduce heat to medium. Add remaining 1 tbs butter. Add sliced mushrooms (about 8 oz) and cook, stirring occasionally, 4 to 6 minutes until browned and most moisture has evaporated.

Add finely chopped shallot and cook 1 to 2 minutes until softened. Add minced garlic and cook 30 seconds until fragrant, taking care not to burn.

Increase heat to medium-high. Pour in 3 tbs Marsala wine to deglaze the pan, scraping up browned bits with a wooden spoon. Let the wine bubble and reduce by about half (about 1 to 2 minutes).

Stir in 3 tbs chicken stock and simmer 1 to 2 minutes to slightly thicken. If the pan is dry, add the stock first then the wine; the goal is a glossy pan sauce.

Reduce heat to low and stir in 1 tbs heavy cream to round the sauce. Taste and season with remaining 1 tsp salt and 1/2 tsp black pepper (adjust to taste). Add 1 tsp fresh thyme leaves if using.

Return the cooked chicken cutlets and any accumulated juices to the skillet, spooning sauce and mushrooms over them. Warm through 1 to 2 minutes, spooning sauce over the chicken to coat.

Sprinkle 1 tbs chopped fresh parsley over the chicken and sauce. Serve immediately.