Combine bulgur with 1/2 cup warm water in a small bowl; let soak until the bulgur is tender and most of the water has been absorbed, 20 to 30 minutes. Drain any excess liquid.
Meanwhile, heat oil in a large nonstick skillet over medium heat. Add onion; cook, stirring often, until the onion softens, about 4 minutes. Add garlic and prunes; cook, stirring often, until fragrant, about 2 minutes. Don't let the garlic burn. Stir in cumin, coriander and allspice; cook, stirring constantly, for 1 minute. Add remaining 2 tablespoons water; cook until it is absorbed, about 1 minute. Remove from the heat.
Combine beef, egg, salt, pepper, the onion-prune mixture and the bulgur in a mixing bowl; mix thoroughly. Shape into four 3/4-inch-thick patties. ightly oil the grill rack (hold a piece of oil-soaked paper towel with tongs and rub it over the grate). Grill the burgers over medium heat, turning once, until browned and cooked through, about 4 minutes per side. Warm pitas on the grill, if desired. Serve the burgers with the pitas.