Ingredients

• 1/2 cup chopped onion, divided
• 1 tablespoon olive oil plus additional for brushing
• 1/2 cup bulgur
• 1 cup water
• 1 cup canned pinto beans, rinsed and drained
• 1 1/2 tablespoon soy sauce
• 3/4 cup walnuts (2 1/2 ounces)
• 2 garlic cloves, coarsely chopped
• 1/2 cup packed cilantro sprigs
• 3/4 teaspoon ground cumin
• 1/4 teaspoon cayenne
• 1/4 cup mayonnaise
• 1/4 teaspoon grated lime zest
• 1/2 teaspoon fresh lime juice
• 4 slices multi-grain bread,toasted
• Equipment: a perforated grill sheet
• Accompaniments: lettuce; sliced tomato

Instructions

1. 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.
2. 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.
3. Form rounded 1/2 cups of mixture into 4 (31/2-inch-diameter) patties. Chill at least 10 minutes.
4. While patties chill, stir together mayonnaise, zest, and juice.
5. Prepare grill for direct-heat cooking over medium-hot charcoal (medium heat for gas). Put perforated grill sheet on grill and preheat 10 minutes.
6. Brush patties all over with oil.
7. 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.
8. Serve burgers open-faced on toast with lime mayonnaise.