Season both sides of the chicken breasts with sea salt and freshly ground black pepper.
Place all-purpose flour in a shallow bowl and dredge each chicken breast, shaking off any excess flour.
Heat a large skillet over medium-high heat and melt 2 tablespoons of the butter with 3 tablespoons of olive oil until sizzling.
Add two pieces of the dredged chicken to the pan and cook 5–8 minutes per side, until golden and the internal temperature is 165°F (74°C). Remove to a plate and repeat with remaining chicken if needed.
Leave the pan juices and oil in the skillet, then add the lemon juice, chicken stock, and rinsed capers. Bring to a boil, scraping up any browned bits from the bottom of the pan.
Return the cooked chicken to the skillet and simmer in the sauce for about 5 minutes to heat through and meld flavors; then remove the chicken to a platter.
Add the remaining 2 tablespoons of butter to the skillet and whisk vigorously until the sauce is smooth and slightly thickened.
Pour the sauce over the chicken and sprinkle with chopped parsley before serving.