Trim all visible fat and tendons from the chicken breasts and, if needed, trim the underside so pieces are an even thickness.
Make small crosswise slits down each breast to help the marinade penetrate.
Finely chop the capers with a chef's knife and crush the dried oregano with a rolling pin or meat mallet.
In a mixing bowl, whisk together the olive oil, lemon juice, crushed oregano, garlic puree, caper juice, chopped capers, and fresh ground black pepper until combined.
Place the chicken breasts in a resealable plastic bag or a shallow dish and pour the marinade over them, ensuring each piece is coated.
Refrigerate and marinate for preferably 4–6 hours or longer for more flavor.
When ready to cook, drain the chicken in a colander and discard excess marinade.
Preheat a grill or stovetop grill pan to medium-high and lightly spray or oil the grates.
Place the chicken on the grill at an angle to the grates and cook undisturbed for 3–4 minutes to develop grill marks.
Check the marks, then rotate each piece 90 degrees and cook another 3–4 minutes on the same side.
Flip the chicken and cook the second side for 2–3 minutes more, or longer for very thick breasts, until the meat feels firm but not hard when pressed.
Remove from heat and serve hot.