Butterfly each chicken breast horizontally, then place between plastic wrap and pound to an even ¼-inch thickness. Pat completely dry with paper towels.

In a shallow dish, combine flour, salt, and pepper. Lightly coat each chicken breast in the flour mixture, shaking off excess. Place on a wire rack for 5 minutes to allow the coating to set.

Heat a 12-inch stainless steel skillet over medium-high heat until a droplet of water skitters across the surface. Add olive oil and 1 tablespoon of butter, swirling to coat the pan.

Once the fat is shimmering but not smoking (approximately 350°F), add the chicken breasts to the pan without overcrowding. Cook undisturbed for 3-4 minutes until golden brown. Flip once and cook an additional 2-3 minutes until the internal temperature reaches 155°F. Transfer to a clean plate and tent with foil.

Reduce heat to medium and add shallots to the same pan. Cook for 30 seconds until fragrant but not browned. Add garlic and cook for an additional 15 seconds, stirring constantly to prevent burning.

Add white wine, using a wooden spoon to scrape up all browned bits from the bottom of the pan. Simmer until the wine is reduced by half, about 2-3 minutes.

Add chicken broth, lemon juice, and capers. Bring to a simmer and reduce by one-third, approximately 5-7 minutes, or until the sauce can lightly coat the back of a spoon.

Reduce heat to low and whisk in the remaining 2 tablespoons of cold butter, one cube at a time, until fully emulsified. The sauce should appear glossy and slightly thickened.

Place chicken and any accumulated juices back into the pan, gently turning to coat with sauce. Simmer for 1-2 minutes until the chicken reaches a final internal temperature of 165°F.

Remove from heat, sprinkle with fresh parsley, and garnish with lemon slices. Serve immediately, spooning additional sauce over the chicken.