Set 4 whole garlic cloves in a small skillet over medium-high heat; toast until spotty brown, about 5 minutes. Peel, mince and set aside.
Break up lamb in a medium bowl. Sprinkle roasted garlic, feta, oregano, salt and pepper over meat; stir with a fork to combine. Divide into 4 portions. Toss meat back and forth between cupped hands to form a ball. Pat with fingertips to flatten to about 4 inches wide. Refrigerate.
Mix cucumber, sour cream, mint, vinegar, 1 minced clove of garlic, and salt and pepper to taste in a small bowl. Refrigerate sauce until ready to serve.
Build a hot fire on one side of grill. Replace rack. When coals are covered with white ash, add burgers. Cover and cook, turning only once, until done, about 4 to 5 minutes per side for medium to medium-well burgers.
Grill pitas until spotty brown and warm, a minute or so per side. Split crosswise. Spread a couple of tablespoons of sauce over 4 pita halves. Top each with an onion slice, a tomato slice and a burger. Spread a portion of the remaining sauce over each burger. Top with remaining pita halves.