Prepare a medium charcoal or gas grill fire. Clean and oil the grill grates. Gently mix the lamb with the paprika, oregano, garlic, and 1 tsp. salt. Form into four 1/2-inch-thick patties.
On a large plate, lay out the slices of feta and sprinkle with the olive oil and 1 Tbs. of the dill. In a small bowl, toss the onion with the vinegar, sugar, 1/2 tsp. salt, and the remaining 1 Tbs. dill, and let sit for10 to 15 minutes at room temperature.
Grill the burgers on one side until they have good grill marks, about 5 minutes. Flip and cook the other side until it has good grill marks, too, and the burgers are just light pink inside (make a nick with a paring knife), about 5 minutes for medium doneness.
Serve the burgers in the warmed pita with the feta, tomato, cucumber, and a heaping teaspoon of pickled onions.