Slice the chicken breasts in half horizontally to create thin cutlets. Place them between two sheets of plastic wrap and gently pound them to an even 1/2-inch thickness.

In a shallow dish, whisk together the flour, kosher salt, and black pepper. Dredge each chicken cutlet in the flour mixture, shaking off any excess.

Heat the olive oil and 2 tablespoons of the unsalted butter in a large skillet over medium-high heat. Add the chicken cutlets in batches, cooking for 3 to 4 minutes per side until golden brown and cooked through. Transfer the cooked chicken to a plate and cover loosely with foil to keep warm.

In the same skillet, add the chicken broth, dry white wine, and fresh lemon juice. Bring to a simmer, using a wooden spoon to scrape up any browned bits from the bottom of the pan. Let the liquid reduce for about 2 to 3 minutes.

Stir in the drained capers and whisk in the remaining 2 tablespoons of butter until the sauce is smooth and slightly thickened.

Return the chicken cutlets to the skillet and spoon the warm lemon-caper sauce over them. Garnish with chopped fresh parsley and lemon slices, and serve immediately.