In a medium bowl, combine chicken, garlic, and oregano sprigs. Add lemon quarters, squeezing juice into bowl as you go; season with salt and pepper. Cover with plastic wrap and refrigerate 30 minutes.
Set a medium pot of salted water to boil. Heat grill pan or grill to medium-high. Clean and lightly oil hot grill. Remove chicken from marinade and grill until cooked through, about 3 minutes total, flipping once. Transfer cutlets to a platter and tent with foil to keep warm.
Meanwhile, cook orzo according to package instructions. Drain orzo and transfer to a medium bowl. Stir in chopped oregano, 1 tablespoon lemon juice, oil, and hazelnuts; season with salt and pepper. Serve chicken and orzo with lemon wedges if desired.