Slice each pounded chicken breast in half, making 2 portions each. You should have 4 total chicken cutlets. You can also use 4 boneless chicken thighs.
In a large bowl, combine chicken with olive oil, salt, Italian seasoning, garlic powder, onion powder, and black pepper. Toss well to coat chicken in seasoning. Cover bowl with plastic wrap and refrigerate until ready to cook, up to 1 day.
When ready to cook, preheat gas grill to medium-high heat. Cook chicken 4-5 minutes per side, until chicken juices run clear.
Remove chicken from grill and place on a platter.
For the sandwiches:
Toast hamburger bun halves for 1-2 minutes on the hot grill, if desired. Top buns with lettuce, avocado slices, grilled chicken, and additional Parmesan.
Pour creamy Caesar dressing over the chicken sandwiches (1/4 cup each, more or less, to taste).