Pat 1 pound chicken cutlets, trimmed, dry with paper towels and season with ½ teaspoon table salt and ½ teaspoon pepper; set aside.
Whisk 3 tablespoons red wine vinegar, 2 teaspoons Dijon mustard, 1 teaspoon mayonnaise, remaining ½ teaspoon salt, and remaining ½ teaspoon pepper in large bowl until mixture is milky in appearance and no lumps of mayonnaise remain.
Whisking constantly, slowly drizzle in ⅓ cup extra-virgin olive oil until glossy and lightly thickened; set aside.
Heat remaining 1 tablespoon oil in 12-inch nonstick skillet over medium-high heat until just smoking.
Add chicken and cook until golden brown and meat registers 160 degrees, 3 to 4 minutes per side.
Transfer chicken to carving board; let rest 5 minutes.
Cut chicken into ½-inch pieces.
Rewhisk dressing to combine and add 1 rinsed can chickpeas; 1 English cucumber, quartered lengthwise and cut into ¼-inch pieces; 10 ounces halved cherry tomatoes; 8 ounces cherry-size mozzarella balls, halved; ½ red onion, chopped fine; ½ cup parsley leaves; ½ cup jarred sliced banana peppers, chopped coarse; and ½ cup pitted kalamata olives, halved.
Transfer to serving platter and top with chicken.
Season with salt and pepper to taste. Serve.
