Season both sides of the chicken with 1½ teaspoons of sea salt and the black pepper. Place the flour in a shallow dish, dredging each piece of chicken until well-coated, shaking off any excess. Discard remaining flour after coating all pieces.

In a large skillet over medium-high heat, melt 3 tablespoons of butter with 2 tablespoons of olive oil. Add half of the chicken and cook for 2-3 minutes on each side, until golden brown and thoroughly cooked. Remove the cooked chicken and repeat with the remaining pieces, adding more butter and oil as needed. Transfer the browned chicken to a serving plate.

To the same skillet, add the minced shallot and cook, stirring occasionally, for approximately 2 minutes, until softened and aromatic. Incorporate the garlic, cooking for an additional minute. Pour in the chicken stock and let it simmer until reduced by half, about 4-5 minutes.

Lower the heat and stir in 4 tablespoons of butter, the drained capers, lemon zest, lemon juice, and the remaining ½ teaspoon of salt. Once the butter has melted and the sauce has blended, pour the mixture over the chicken. Garnish with freshly chopped parsley if desired before serving.