Pound chicken cutlets between sheets of plastic wrap to a uniform 0.25 inch thickness using a meat mallet.
Combine flour, salt, and pepper in a shallow dish; dredge each cutlet and shake off excess flour.
Heat olive oil and 1 tablespoon of butter in a large heavy skillet over medium-high heat until shimmering.
Sear chicken cutlets in batches for 2 to 3 minutes per side until golden brown and cooked through; remove and set aside on a warm plate.
Add sliced mushrooms to the skillet and sauté for 5 minutes until browned and moisture has evaporated.
Deglaze the skillet with white wine, scraping the bottom to release the fond, and simmer until liquid is reduced by half.
Stir in lemon juice and capers, then whisk in the remaining 3 tablespoons of cold butter to emulsify the sauce.
Return chicken to the skillet briefly to coat with the sauce and garnish with chopped parsley before serving.