With a chicken breast flat on the cutting board, make a horizontal cut into the fat side of the breast and slice the breast horizontally in half to make 2 cutlets. Repeat with the second breast. Cover with a layer of plastic wrap and use a meat mallet or a small heavy skillet to lightly pound the cutlet to an even ½-inch thickness. Season both sides of the chicken with ½ teaspoon of the salt and ¼ teaspoon black pepper.

Heat the olive oil in a 10-inch skillet with about 2-inch sides or a large Dutch oven over medium heat. Add the chicken, in batches if necessary, and cook until cooked through, 2 to 3 minutes per side. Remove the chicken to a plate and set aside.

Reduce the heat to medium-low and add the butter, shallots, and remaining 1 teaspoon salt and cook, stirring, until lightly golden, 1 to 2 minutes. Add the garlic and cook, stirring frequently, until golden and lightly fragrant, 30 seconds to 1 minute, taking care not to burn the garlic. Increase the heat to medium, stir in the mushrooms and thyme, and cook until the mushrooms are soft, 6 to 8 minutes. Stir in the vinegar and cook for 1 minute, stirring occasionally.

Reduce the heat to medium and sprinkle in the flour. With a wooden spoon, cook, stirring frequently for 30 seconds to 1 minute to lightly toast the flour. Add the chicken broth, half-and-half, and Parmesan and stir to combine. Bring the sauce to a simmer and cook until the sauce has thickened, 1 to 2 minutes. Stir in the lemon juice, return the chicken and any juices to the pan, and gently rewarm the chicken for 30 seconds to 1 minute.

Transfer to a platter, top with a few grinds of pepper, and garnish with the parsley. Serve with lemon quarters and a big slice of crusty bread or on top of Perfect Stovetop Rice.