Prepare Chicken: Slice chicken breasts horizontally to create 8 thin cutlets. Place them between two sheets of plastic wrap and pound to a uniform ¼-inch thickness.

Dredge: Mix the flour, salt, and pepper in a shallow dish. Dredge each chicken cutlet lightly in the flour mixture, shaking off any excess. The thin coating is key.

Sear Chicken: Heat olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Sear the chicken cutlets in batches, cooking for 2-3 minutes per side until golden brown and cooked through. Remove the chicken to a platter and cover to keep warm.

Deglaze Pan: Reduce the heat to medium. Pour the wine or broth into the skillet and scrape up all the browned bits (fond) from the bottom. Let the liquid reduce by about half, about 2 minutes.

Make the Sauce: Stir in the lemon juice and chicken broth. Bring the mixture to a simmer. Remove the pan from the heat and swirl in the remaining 2 tablespoons of butter, one tablespoon at a time. The butter will melt and emulsify, creating a beautifully **thick, viscous sauce**.

Add Capers: Stir in the drained capers and half of the chopped parsley. Taste the sauce and add salt or pepper if necessary.

Finish and Serve: Return the cooked chicken cutlets to the pan and gently spoon the **gooey, savory sauce** over them to coat. Cook for 1 minute just to reheat. Serve immediately, garnished with the remaining fresh parsley and lemon slices.