Slice the chicken breasts in half horizontally to create 4 thin cutlets. If they are uneven, gently pound them to a uniform 1/4-inch thickness.

In a wide, shallow dish, whisk together the all-purpose flour, kosher salt, and black pepper. Lightly dredge each chicken cutlet in the flour mixture, shaking off any excess.

Heat the olive oil and 1 tablespoon of the butter in a large skillet over medium-high heat until the butter is melted and bubbling.

Place the chicken cutlets in the skillet, working in batches if necessary so as not to crowd the pan. Sear for 3 to 4 minutes per side, until golden brown and cooked through. Transfer the chicken to a plate and cover loosely with foil.

Reduce the skillet heat to medium. Add the minced garlic and sauté for 30 seconds until fragrant.

Pour the dry white wine into the skillet to deglaze, using a wooden spoon to scrape up any browned bits from the bottom. Let the wine simmer and reduce by half, which should take about 1 to 2 minutes.

Stir in the chicken broth, fresh lemon juice, and drained capers. Allow the sauce to simmer for another 2 to 3 minutes until it slightly thickens.

Remove the skillet from the heat. Add the remaining 2 tablespoons of butter and whisk continuously until the butter is melted and the sauce emulsifies into a smooth, silky texture.

Return the warm chicken cutlets to the skillet. Spoon the lemon caper sauce generously over the top of the chicken.

Garnish with chopped fresh parsley and serve immediately.