Season the chicken all over with the pepper and 1½ teaspoons of the salt. Place the flour in a shallow bowl. Dip each piece of the chicken in the flour, turning to coat and shake off any excess. Discard the flour when finished dredging.

Heat 3 tablespoons butter and 2 tablespoons oil in a large skillet set over medium-high until the butter has melted. Add half of the chicken and cook until golden brown and cooked through, about 2-3 minutes per side. Repeat with the remaining chicken, adding more butter and oil to the skillet as needed. Transfer the chicken to a serving platter.

Add the shallot to the pan and cook, stirring occasionally, until soft and fragrant, about 2 minutes. Add the garlic and cook, stirring, until fragrant, 1 minute more. Stir in the stock and simmer until the liquid is reduced by half, 4-5 minutes. Reduce heat to low, then stir in 4 tablespoons of the butter, capers, lemon zest, lemon juice and the remaining 1/2 teaspoon of salt.

Pour the sauce over the chicken and garnish with parsley before serving.