Mix 1 garlic clove, finely grated, ⅓ cup red wine vinegar, 2 Tbsp. caper brine, and 2 Tbsp. extra-virgin olive oil in a large bowl to combine. Slice 2 large skinless, boneless chicken breasts (about 1 lb. total) in half like you’re going to butterfly them, but cut all the way through to make 4 evenly thick cutlets (⅓" thick or less). Add to bowl with marinade and turn to coat. Let sit at least 15 minutes and up to 1 hour.

Heat remaining 2 Tbsp. extra-virgin olive oil in a large skillet over medium-high. Remove chicken from marinade; pat dry (no need to salt). Set marinade aside. Arrange chicken in pan; cook until golden, about 3 minutes. Turn and cook on other side until golden and cooked through, about 2 minutes. Transfer chicken to a platter; reserve skillet.

Stir 1 tsp. honey and ¼ cup water into reserved marinade, then pour into reserved skillet. Cook, scraping up brown bits, 1 minute. Add 3 Tbsp. unsalted butter, cut into pieces, and 1 Tbsp. drained capers; season with freshly ground pepper. Cook, stirring, until butter is melted and sauce is emulsified, 1–2 minutes. Drizzle sauce over chicken; top with coarsely chopped parsley.