Preheat the grill: Set the grill to medium-high heat (375-400°F). Lightly brush the grates with 1 tablespoon olive oil to prevent sticking.
Season the chicken: Pat the chicken dry and season both sides with black pepper and 1½ teaspoons salt. Lightly coat with 1 tablespoon olive oil.
Grill the chicken: Place on the grill and cook for 3-4 minutes per side, or until the internal temperature reaches 165°F.
Rest the chicken: Transfer to a serving platter and loosely tent with foil to keep warm.
Sauté the aromatics: In a 12-inch deep skillet over medium heat, melt 2 tablespoons butter with 1 tablespoon olive oil. Add the shallot and cook for 2 minutes until soft. Stir in the garlic and cook for 1 minute more, until fragrant.
Simmer the sauce: Pour in the chicken stock and bring to a simmer. Cook until the liquid reduces by half (about 4-5 minutes).
Finish with butter & lemon: Lower the heat and stir in the remaining 2 tablespoons butter, capers, lemon zest, and lemon juice. Stir continuously until the butter is fully melted and the sauce is smooth. Avoid boiling to prevent separation.
Coat the chicken: Pour the piccata sauce over the grilled chicken, making sure each piece is well-coated.
Garnish & serve: Sprinkle with fresh parsley and extra lemon zest, if desired. Serve immediately with pasta, rice, grilled vegetables, or a fresh salad, drizzling extra sauce over the top.