Cook half of onion with 1/4 teaspoon salt in oil in a small heavy saucepan over medium heat, stirring occasionally, until golden, 5 to 7 minutes. Add bulgur and water and cook, covered, over low heat until water is absorbed, 15 to 18 minutes. Transfer to a bowl and stir in beans and soy sauce.
Pulse bulgur mixture, walnuts, garlic, cilantro, cumin, cayenne, a rounded 1/4 teaspoon salt, 1/2 teaspoon pepper, and remaining onion in a food processor until finely chopped.
Form rounded 1/2 cups of mixture into 4 (31/2-inch-diameter) patties. Chill at least 10 minutes.
While patties chill, stir together mayonnaise, zest, and juice.
Prepare grill for direct-heat cooking over medium-hot charcoal (medium heat for gas). Put perforated grill sheet on grill and preheat 10 minutes.
Brush patties all over with oil.
Oil grill sheet, then grill burgers on grill sheet, covered only if using a gas grill, carefully turning once, until golden brown, about 8 minutes total.
Serve burgers open-faced on toast with lime mayonnaise.
Cooks' notes: ·Burgers can be cooked on the stove. Heat 1 tablespoon oil in a 12-inch heavy skillet over medium heat until it shimmers, then cook patties, carefully turning once, until golden brown, about 8 minutes total. ·Uncooked patties can be chilled, covered, up to 4 hours.
Per serving: 399 calories, 22g fat (3g saturated), 4mg cholesterol, 768mg sodium, 43g carbohydrates, 10g fiber, 12g protein Nutritional analysis provided by Nutrition Data See Nutrition Data's complete analysis of this recipe ›