Put the lamb in bowl. Using a box grater, grate the onion over the bowl so that the pulp of the onion and juices fall into the meat. Add the spices, chile pepper, egg, bread crumbs, salt and pepper, to taste, and mix to combine. Form the meat mixture into 4 patties, thinner at the middle and thicker at the edges for even cooking.
Heat a large skillet over medium-high heat and coat with oil, or coat the burgers in oil and grill on outdoor grill. Cook the patties about 10 minutes, turning once.
While the patties cook, add the yogurt to a small bowl. Grate the garlic into the yogurt using a fine grater or zester. Add the mint and stir to combine.
Char the pitas over an open gas burner or outdoor grill.
Eat the burgers in or on the pita bread topped with garlic-mint yogurt, sliced red onion, and some shredded cabbage.