Prepare chicken: Slice each chicken breast horizontally into 4 thin cuts, rub with 2 tsp olive oil, and season with salt and pepper.
Cook chicken: Heat half of the remaining olive oil in a non-stick frying pan. Fry the chicken over medium heat for 3 minutes on each side or until fully cooked. Transfer to a plate.
Sauté onions: Using the same pan, add the remaining oil and sauté the onions for 5 minutes until softened.
Assemble salad: Slice the cooked chicken, collecting any juices. In a large serving bowl, layer the chicken with the sautéed onions, salad leaves, sliced cucumber, and dried cranberries.
Prepare dressing: In a small bowl, whisk together the cranberry sauce, lime juice, 2 tablespoons water, and any juices from the chicken. Drizzle over the salad before serving.
Serve: Serve the cranberry chicken salad fresh with the tangy lime dressing.