Bulgur Veggie Burgers with Lime Mayonnaise

Name: Bulgur Veggie Burgers with Lime Mayonnaise
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