Prep the Chicken: Place chicken breasts between two sheets of plastic wrap or parchment paper. Use a meat mallet to gently pound them to an even thickness (about 1/2 inch) for even cooking.
Season and Dredge: In a shallow dish, mix the flour, salt, and pepper. Dredge each chicken breast in the flour mixture, coating both sides and shaking off excess.
Cook the Chicken: In a large skillet, heat the olive oil and 2 tablespoons of butter over medium-high heat. Once hot, add the chicken breasts to the skillet. Sear for 4-5 minutes on each side until golden brown and cooked through. Remove chicken from the skillet and set aside.
Make the Sauce: In the same skillet, reduce the heat to medium. Add the minced garlic and sauté for about 30 seconds until fragrant. Carefully pour in the chicken broth and lemon juice, scraping up any brown bits from the bottom of the skillet.
Add Capers: Stir in the capers and the remaining butter. Allow the sauce to simmer for 3-4 minutes to reduce slightly.
Combine and Serve: Return the cooked chicken to the skillet, spooning sauce over the top. Simmer for an additional 2-3 minutes until heated through.
Garnish and Plate: Plate each chicken breast and pour the luscious lemon-caper sauce over it. Sprinkle freshly chopped parsley on top for a burst of color.