For burgers
1 cup boiling-hot water
1/3 cup bulgur
3/4 teaspoon salt
1 medium onion, quartered
1/4 cup packed fresh cilantro leaves
1/4 cup packed fresh flat-leaf parsley leaves
1 lb ground lamb
1/2 teaspoon paprika (not hot)
1/4 teaspoon ground allspice
1/4 teaspoon black pepper
1/4 teaspoon cayenne

For sauce
1 small garlic clove
1/8 teaspoon salt
1/2 cup walnuts (1 1/2 oz)
1/4 cup water
1 teaspoon fresh lemon juice
1/8 teaspoon cayenne

For pitas
4 (4-inch) pita loaves
1 tablespoon olive oil
Paprika (not hot) for dusting

Garnish: fresh cilantro sprigs

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.