Soak bulgur for burgers: Pour boiling-hot water over bulgur with 1/2 teaspoon salt in a small bowl and soak 15 minutes, then drain in a sieve.
Make sauce while bulgur soaks: Mince garlic and mash to a paste with salt (1/8 teaspoon) using a large heavy knife, then transfer to a food processor and blend with remaining sauce ingredients until smooth. Transfer to a bowl.
Prepare burgers: Preheat broiler.
Pulse onion and herbs in cleaned processor until finely chopped, then transfer to a bowl and stir in bulgur, lamb, paprika, allspice, pepper, cayenne, and remaining 1/4 teaspoon salt until just combined. Form lamb mixture into 4 patties (4 inches in diameter).
Broil pitas and burgers: Arrange pitas on a baking sheet. Brush with oil, then lightly dust with paprika and season with pepper. Broil on lowest rack until toasted, 1 to 3 minutes (watch to prevent burning). Keep pitas warm, loosely covered with foil, while broiling burgers.
Oil rack of a broiler pan and heat 4 inches from heat until hot, 3 to 5 minutes.
Broil burgers, turning over once, until cooked through, 5 to 7 minutes total. Serve burgers on pitas, topped with sauce.