If chicken breasts are thick, slice each breast horizontally to make 2 thinner cutlets or butterfly and then place between two sheets of plastic wrap. Gently pound with a meat mallet or rolling pin to about 1/4 inch thickness so pieces are even.

Season

Pat cutlets dry with paper towels. Season both sides evenly with 1 tsp salt and 1/2 tsp black pepper (reserve remaining salt and pepper for dredging).

Prepare dredging station

Place flour in a shallow dish and season with remaining 1/2 tsp salt and 1/2 tsp black pepper. In a second shallow bowl, whisk together the eggs and 2 tbsp milk until combined.

Dredge and shake off excess

Working one at a time, dredge each cutlet in the seasoned flour, shaking off excess, then dip into the egg wash, letting excess drip back into the bowl, then lightly return to the flour for a thin coating. Set on a plate while you heat the pan.

Brown chicken

In a large skillet over medium-high heat, add 3 tbsp vegetable oil and 3 tbsp unsalted butter. When butter is foaming and oil is shimmering, add cutlets (work in batches if needed; do not crowd). Cook 2–3 minutes per side until golden brown and just cooked through (internal temp 160–165°F / 71–74°C). Transfer cooked cutlets to a warm plate and tent loosely with foil; reserve pan with brown bits for the sauce.

Make Sauce & Finish

Sauté aromatics

Reduce heat to medium. Add 1 tbsp extra-virgin olive oil to the same skillet. If there is excess fat, remove all but about 1 tbsp. Add minced shallot and cook 1–2 minutes until softened. Add minced garlic and cook 30 seconds until fragrant, stirring so it doesn't burn.

Deglaze with wine

Carefully add 3 tbsp dry white wine to the pan to deglaze, scraping up browned bits with a wooden spoon. Let the wine reduce by about half, 1–2 minutes.

Add broth, lemon, and capers

Stir in 3 tbsp chicken broth, 3 tbsp fresh lemon juice, and 2 tbsp drained capers. Bring to a gentle simmer and cook 1–2 minutes to meld flavors.

Mount with butter

Reduce heat to low. Add 2 tbsp cold unsalted butter, one small cube at a time, whisking constantly to emulsify the sauce into a glossy, slightly thickened consistency. Taste and adjust seasoning with up to 1/4 tsp more salt and a pinch of black pepper if needed.

Return chicken to pan

Return the browned chicken cutlets and any accumulated juices to the skillet, spooning sauce over them. Warm through for 1–2 minutes (do not simmer vigorously or the butter will separate).

Finish and garnish

Turn off heat. Stir in 2 tbsp chopped fresh parsley and optional 1 tsp lemon zest. Spoon additional sauce and capers over each cutlet before serving.