Line a strainer with four layers of cheesecloth or one coffee filter and place over a bowl. Place yogurt in strainer; cover yogurt with edges of cheesecloth. Refrigerate for 2 hours.
Remove yogurt from cheesecloth and discard liquid from bowl. Pat cucumber dry with paper towels. In a small bowl, combine the yogurt, cucumber and remaining sauce ingredients. Cover and chill until serving.
In a small bowl, combine the butter, basil and garlic. Spread over cut sides of bun; set aside.
For burgers, in a large bowl, combine the first seven ingredients. Crumble lamb over mixture and mix well. Shape into six patties.
Grill patties, covered, over medium heat or broil 4 in. from the heat for 5-7 minutes on each side or until a meat thermometer reads 160° and juices run clear.
Grill buns, cut sides down, over medium heat for 30-60 seconds or until lightly toasted. Serve burgers on buns with avocado, tzatziki sauce and, if desired, lettuce, onion and tomato. Yield: 6 servings.