Preheat barbecue to medium-high. In a large bowl, mix lamb, eggs, onion, garlic, mint, lemon zest and juice. Season with salt and pepper. Place a sheet of foil onto grates to cover most of surface.
Shape lamb mixture into equal sausage-shaped logs (it should make 6 koftas at 7 in./18 cm). Push a skewer through the length of each kofta, squeezing to compact around skewer.
Brush warm foil with oil. Reduce barbecue to medium and place skewers on foil to cook, 2 to 3 min. on each of the 4 sides. Meanwhile, wrap pita in foil and place on upper rack to warm through, 1 to 2 min.
Slide each kofta off skewer into a pita. Garnish with lettuce, tzatziki, cucumber and red pepper.