Lay chicken flat; holding a sharp knife parallel to work surface, split chicken in half horizontally. Cover with plastic wrap; pound each cutlet with the flat side of a meat mallet or the bottom of a small pan until inch thick.
In a shallow dish, combine 2 tablespoons oil and 1 tablespoon lemon juice; season with salt and pepper. Add chicken; turn to coat. Heat a large skillet over medium-high; lightly brush with oil. Cook chicken until opaque throughout, 1 to 2 minutes per side.
In a bowl, toss together chickpeas, onion, basil, 2 tablespoons oil, and 1 tablespoon lemon juice; season with salt and pepper.
Top chicken with chickpea relish; serve with arugula and lemon wedges.