Butterfly each chicken breast and cut in half to make thinner cutlets.
Combine 1/2 cup brown sugar, 1/4 cup olive oil, 1 tablespoon Worcestershire, 1 tablespoon soy sauce, and juice of 1/2 lemon in a large resealable bag; add chicken, seal, and marinate in the refrigerator for 30 minutes up to 2 hours.
Make the dressing by whisking together 1/2 cup mayonnaise, 2 tablespoons Greek yogurt, 1 teaspoon Dijon mustard, 1/2 teaspoon Worcestershire, 1/8 teaspoon anchovy paste, juice of 1/2 lemon, and 1–2 cloves minced garlic; refrigerate until ready.
Make croutons: cut bread into bite-size pieces, then fry in a skillet over medium heat with 1 tablespoon butter and 1 minced clove garlic, stirring, until golden, about 4–6 minutes; transfer to a plate to cool.
Preheat and oil the grill grate (or heat a grill pan) to high, then reduce to medium-high before cooking.
Place chicken on the grill and brush the tops with BBQ sauce; cook about 4 minutes, turn, brush the other side with BBQ sauce, and cook another 4 minutes or until cooked through and no longer pink inside.
While the chicken cooks, tear the romaine, slice the avocado, and shave the Parmesan.
Slice the cooked chicken and assemble the salad by tossing lettuce, croutons, avocado, and Parmesan with the dressing, then top with sliced BBQ chicken.