Preheat a grill pan over medium heat and lightly coat with cooking spray. Sprinkle the chicken with 1/4 teaspoon each salt and pepper. Grill until well marked and cooked through, 3 to 5 minutes per side; transfer to a plate. Add the bread to the pan and grill until toasted, 2 to 3 minutes per side. Transfer to a plate and rub with the garlic.
Make the dressing: Puree the olive oil, vinegar, 2 tablespoons water, half each of the cucumber, tomatoes, parsley and almonds, 1/2 teaspoon salt, and pepper to taste in a blender until smooth.
Cut the chicken into strips. Tear the bread into bite-size pieces. Combine the arugula, chickpeas, chicken, bread and the remaining cucumber, tomatoes, parsley and almonds in a large bowl. Add half of the dressing, season with salt and pepper and toss. Divide among shallow bowls. Drizzle with the remaining dressing.