Step 1: Make the burgers: Place the lamb, onion, garlic, mint, hot pepper, and cumin in a mixing bowl and stir to mix with a wooden spoon.
Step 2: Form the lamb into 4 patties, each about 1 inch thick. Work with a light touch, handling the meat as little as possible. Make a slight depression in the center—the patty should be slightly concave. You can form the burgers up to 6 hours ahead. Arrange them on a plate lined with plastic wrap and cover with more plastic wrap. Store in the refrigerator until grilling.
Step 3: Just before cooking, generously season the burgers on both sides with salt and pepper, turning gently.
Step 4: Set up your grill for direct grilling and preheat to high. Ideally, you’ll grill over a wood fire or over charcoal or gas with a handful of soaked hardwood chips. Brush and oil the grill grate. Arrange the burgers on the grate. Grill until the bottoms are browned, about 3 minutes. Gently invert the burgers, using a spatula. However, never press the burgers with the spatula—you’ll force out the succulent juices. Continue grilling the burgers until cooked to taste, 5 minutes in all for rare, 7 minutes for medium-rare, 8 to 9 minutes for medium. (Note: unless you’re absolutely sure about the purity of your meat, cook the burgers to medium—160 degrees.) Use an instant-read meat thermometer inserted through the side of the burger to verify the internal temperature: medium should read 160 degrees.
Step 5: Meanwhile, cut a slit in the top of the pita breads. Warm the pita on the grill. Insert the burgers, cucumber, tomato, onion, yogurt, and/or cheese and serve at once.