Using kitchen shears, cut chicken down each side of backbone; remove backbone and wing tips. Place breast side up on rimmed baking sheet; press firmly on breastbone to flatten.

Using fingers, gently loosen skin from breasts and legs to form pockets, leaving skin attached at edges.

In small bowl, combine garlic, dill, oregano, half of the oil and the lemon juice; stuff two-thirds under chicken skin, patting to flatten slightly. Slide 6 of the lemon slices under skin. Brush with remaining oil. Turn chicken over: spread with remaining garlic mixture. Top with remaining lemon. Refrigerate for 30 minutes. (Make-ahead: Cover and refrigerate for up to 12 hours.)

Heat 1 burner of 2-burner barbecue or 2 outside burners of 3-burner barbecue to medium. Sprinkle chicken with salt and pepper. Brush grill over unlit burner with oil. Place chicken, skin side down, on oiled grill. Close lid and grill until thermometer inserted in thigh registers 185°F (85°C), about 1 hour. Remove to cutting board; tent with foil and let stand for 10 minutes before cutting with clean kitchen shears into eight pieces.

Line colander with cheesecloth and place over bowl; spoon in yogurt. Refrigerate for 1 hour.

Meanwhile, peel half of the cucumber; seed and dice entire cucumber. In separate colander, set over bowl, stir cucumber with salt; let drain for 30 minutes.

Stir together drained yogurt, cucumber, onion, dill and garlic. Serve with chicken.