Slice chicken breasts in half crosswise into 4 cutlets and lightly pound each piece between sheets of plastic wrap until an even thickness (about ½" thick or less). Season lightly with salt. Place flour in a medium shallow bowl. Place cutlets in bowl and toss to coat in flour. Knock off excess flour and transfer to a plate.

Heat 2 Tbsp. oil in a large skillet over medium-high. Cook cutlets, without moving them, until deeply browned underneath, about 2 minutes. Turn over and cook on the other side just until chicken is nearly cooked. Transfer to a clean plate.

Add garlic and remaining 1 Tbsp. oil to the skillet and cook until golden brown, about 2 minutes. Add wine and capers and reduce until liquid is almost completely evaporated. Add ½ cup water, followed by the butter. Swirl pan vigorously while butter melts to help it form an emulsion with water, about 1 minute.

Return chicken to skillet and simmer until chicken is cooked through and sauce is thick enough to coat a spoon, about 2 minutes. Remove from heat and stir lemon juice into sauce; season with salt. Transfer chicken and sauce to a platter and top with parsley; serve with lemon wedges.