Carefully slice each chicken breast horizontally to produce six thin cutlets. Season both sides generously with salt and black pepper.

Place the flour in a shallow dish and dredge each cutlet lightly, shaking off any surplus. Set the floured cutlets aside in a single layer.

In a large skillet set over medium-high heat, combine 2 tablespoons of olive oil with 2 tablespoons of butter. Once the butter begins to foam, arrange three cutlets in the pan. Sauté each side for approximately 2 to 4 minutes, or until golden and cooked through (internal temperature should reach 165°F). Transfer the cooked chicken to a serving dish. Repeat with the remaining cutlets, adding an additional tablespoon of oil and butter as needed.

With the skillet still over medium-high heat, deglaze the pan by adding the white wine, followed by the lemon juice, 1 tablespoon of butter, and the rinsed capers. Simmer for about 1 minute, scraping up any browned bits to incorporate into the sauce.

Submerge the cooked chicken cutlets in the simmering sauce individually to ensure even coating. Return them to the serving dish and allow the sauce to reduce slightly for 2 additional minutes.