In a food processor, grind the olives, chile, single garlic clove, capers, 1 teaspoon black pepper, fresh oregano, and 1/2 cup olive oil together. Cover and set aside.
Preheat oven to 500 degrees F.
Make sure the lamb leg is cleaned well and any hard fat is trimmed off. Make 8 slits in the leg and stuff them with the rosemary sprigs and remaining sliced garlic. Rub the leg with the remaining olive oil, dry oregano, lemon, salt, and pepper. Roast for about 30 minutes. Lower heat to 375 degrees. Add water to the pan if needed. Cook until the internal temperature of the lamb is 120 to 135 degrees F, about 3 1/2 hours. Remove from the pan and let rest for 10 minutes. Skim the fat from the pan juices and pour into a bowl. Use for another time, this juice can be frozen. Slice the lamb as thinly as possible.
Toast the pita bread and spread the goat cheese on the inside of the pita. Pile in about 4 to 5 slices of the lamb and top with the olive spread and arugula. Serve room temperature.