Whisk together Caesar salad dressing, olive oil, and garlic for the chicken together in a medium bowl. Add chicken, turning to coat. Cover and refrigerate for 4 hours.
When ready to cook, preheat grill. Grease grill grates lightly with oil.
Drizzle the cut side of romaine hearts with 1/2 teaspoon of olive oil each. Sprinkle evenly with salt and pepper for the salad.
Remove chicken from marinade. Add chicken, romaine hearts, and bread slices to the grill. Grill for about 10 minutes, turning the chicken and bread halfway through cooking. You may need to cook the chicken for more time depending on how thick your chicken is. The internal temperature of the chicken should be 165˚F when done.
Roughly chop lettuce and add to a medium bowl. Add Caesar dressing for the salad, avocado, and lemon juice. Stir to combine and set aside.
Chop grilled bread slices into small cubes.
Slice grilled chicken into strips.
To assemble, add chicken to the tortillas, dividing the meat evenly. Top with charred romaine salad, crispy bacon, and shaved parmesan cheese.