Heat 1 tablespoon of olive oil in a large skillet over medium heat. Add the minced garlic and cook for about 30 seconds until fragrant.
Add the ground lamb, oregano, salt, and pepper. Cook for 6 to 8 minutes, breaking up the meat, until it is browned and cooked through. Stir in half the lemon juice and remove from heat.
In a mixing bowl, combine the sliced cucumbers, red onion, chopped dill, and feta cheese. Drizzle with the remaining olive oil and lemon juice, then season with salt and pepper. Toss gently to coat.
Warm the pita breads in a dry skillet or microwave for 20 to 30 seconds until soft and pliable.
Fill each pita with a portion of the seasoned lamb and serve the cucumber salad alongside or tucked inside the wraps.