Heat the large skillet over medium-high heat and add the olive oil.
Add the diced chicken to the skillet, season evenly with salt and pepper, and cook for about 5 minutes, stirring and flipping intermittently until the pieces are about 90% cooked.
Stir in 3 tablespoons balsamic salad dressing and 1 tablespoon balsamic vinegar, tossing to coat the chicken.
Cook for another 1 to 2 minutes over medium heat until the chicken is cooked through, then remove the skillet from the heat and let the chicken rest briefly in the pan.
On a large platter or in a large bowl, arrange the lettuce, diced tomatoes, mozzarella, and diced cucumber.
Add the cooked chicken on top and evenly sprinkle with the minced basil.
Drizzle with additional salad dressing and balsamic vinegar to taste, toss gently to combine, and serve immediately.