How to Make It
Place chicken breasts on a work surface, and cut each horizontally in half to make 2 thin cutlets. Place cutlets between 2 pieces of plastic wrap, and flatten to 1/2-inch thickness using the flat side of a meat mallet (or use a rolling pin or the bottom of a skillet). Season cutlets evenly with 3/4 teaspoon of the salt and 1/2 teaspoon of the pepper. Place 1/2 cup of the flour in a shallow dish; dredge cutlets in flour to coat both sides.
Heat oil in a large cast-iron skillet over medium-high. When oil is shimmering, add chicken cutlets, and cook until golden brown, about 4 minutes. Turn chicken over, and cook until chicken is cooked through, about 2 more minutes. (Cook in 2 batches, if necessary, to avoid crowding in skillet.) Place cutlets on a plate lined with paper towels; cover to keep warm.
Melt butter in skillet over medium; add mushrooms, shallots, and remaining 1/2 teaspoon each of salt and pepper. Cook, stirring often, until mushrooms are browned, 5 to 6 minutes. Add remaining 1 teaspoon flour; cook, stirring often, 1 minute. Add wine, and cook, stirring often, until wine is reduced to about half, about 2 minutes. Stir in whipping cream, parsley, and lemon zest, and remove from heat. Serve sauce with chicken cutlets, sprinkled with lemon zest.