Combine all sauce ingredients in a large measuring cup and set aside. Measure remaining ingredients before starting.
Slice chicken breasts in half lengthwise to create thinner pieces. Cover and pound to about ½-inch thickness.
Season both sides with salt, pepper, and dried parsley.
Heat olive oil in a large skillet over medium-high heat. Sear chicken in batches for 4–5 minutes per side until golden. Remove and set aside.
Turn heat off. Add chicken broth and apple cider vinegar to deglaze the pan. Set heat to medium and scrape up browned bits. Simmer for 2 minutes.
Add garlic and cook for 2–3 minutes until liquid reduces by half.
Stir in butter and flour and cook for 1–2 minutes to remove raw flour taste.
Gradually add the prepared sauce mixture while stirring continuously. Bring to a boil, then reduce to a simmer for about 8 minutes.
Reduce heat to low and slowly stir in Parmesan cheese until smooth. Add lemon juice.
Return chicken to the skillet, spoon sauce over top, and simmer for 5 minutes until sauce thickens. Garnish with parsley and serve.