In a bowl, stir together tomatoes, parsley, 1 tablespoon oil, lemon juice, and capers. Season tomato relish with salt and pepper.
Working with one piece at a time, lay chicken flat; holding a sharp knife parallel to work surface, split in half horizontally. Cover with plastic wrap; lightly pound each cutlet with a meat mallet or bottom of a heavy skillet until 1/4 inch thick.
In a large nonstick skillet, heat 1 tablespoon oil over medium-high. Cook cutlets, in batches, until browned and opaque throughout, 1 to 3 minutes per side. Transfer to a plate, and cover loosely with foil.
To skillet, add spinach; season with salt and pepper. Cook, stirring, until heated through, 2 to 3 minutes. Top chicken with tomato relish, and serve with spinach.
Skip costly cutlets. Make them by pounding whole boneless breasts covered with a layer of plastic wrap.