Place chicken breasts in plastic zippered bag with salsa or picante sauce. Zip shut and refrigerate 1 hour.
Place on barbecue grill and cook for 5 minutes on each side (can be broiled instead). Top with cheese and close top of grill until cheese melts (about 2 minutes). Spread bottoms of each hamburger bun with avocado mixture. Top with lettuce, tomato, chicken breasts and top of buns.