Line a strainer or colander with 4 layers of cheesecloth or 1 coffee filter; place over a bowl. Place yogurt in prepared strainer; cover yogurt with sides of cheesecloth. Refrigerate 2-4 hours. Meanwhile, place cucumber in a colander over a plate; sprinkle with 1/4 teaspoon salt and toss. Let stand 30 minutes.
For burgers, in a large bowl, combine lamb, lemon zest, 2 garlic cloves, oregano, 3/4 teaspoon salt and 1/4 teaspoon pepper, mixing lightly but thoroughly. Shape into ten 1/2-in.-thick patties. Refrigerate 30 minutes.
For sauce, remove yogurt from cheesecloth to a bowl; discard strained liquid. Squeeze cucumber and blot dry with paper towels. Add cucumber, lemon juice, dill, remaining 2 garlic cloves, remaining 1/4 teaspoon salt and remaining 1/8 teaspoon pepper to yogurt, stirring until combined.
Grill burgers, covered, over medium heat until a thermometer reads 160°, 3-4 minutes on each side. Grill buns over medium heat, cut sides down, for 30-60 seconds or until toasted. Serve burgers on buns with lettuce, red onion, feta and sauce.