Combine avocado, tomato, 1 tbsp Worcestershire sauce, oil, lime juice, parsley and 1/4 tsp onion powder. Place in a colander resting over a bowl to allow juices to drain. Reserve juices.
Mix remaining 2 tbsps Worcestershire and 1/4 tsp onion powder with the ground beef; form into 12 thin patties.
Place a spoonful of drained avocado mixture in center of 6 patties; top with remaining 6 patties and seal edges together using a fork. Grill over hot coals or medium heat for about 10 minutes, turning once. Brush with reserved juices.
Insert a cooked pattie inside each pita pocket and stuff with shredded cheese and lettuce.