Place cutlets between sheets of parchment and gently flatten thick areas to about 1/2 inch (1.3 cm). Pat dry. Season with 1/4 teaspoon salt and the pepper. Put flour on a plate, lightly coat the chicken and shake off excess. Keep raw-chicken utensils separate from cooked food.

Heat 1 tablespoon oil and 1 tablespoon butter in a 12-inch skillet over medium-high heat. Cook two cutlets at a time for about 3 minutes per side, adding remaining oil for the second batch if needed. Transfer to a clean plate; the chicken finishes cooking in the sauce.

Add mushrooms to the pan and cook for 6–8 minutes, stirring occasionally, until their moisture evaporates and edges brown. Reduce heat to medium, add shallot and remaining salt and cook 2 minutes. Add garlic for 30 seconds.

Move the pan away from the flame before adding wine. Return to moderate heat and simmer for 3–4 minutes, scraping up browned bits, until reduced by about half. Add broth and simmer for 3–4 minutes, until lightly reduced. Do not flambé.

Return cutlets and collected juices. Simmer gently for 3–5 minutes, or until the thickest part of every piece reaches 165°F (74°C). Add a splash of broth if the sauce reduces too far before the chicken is cooked. Remove from heat and swirl in remaining butter. Sprinkle with parsley and serve.