Cook bulgur according to package directions; drain and set aside to cool. In a food processor, combine cooked bulgur, garbanzo beans, egg white, parsley, 1/4 mint, garlic, cayenne pepper, cumin, salt, and pepper. Form mixture into 6 (approximately 3-inch) patties and place on a large plate. Refrigerate until firm (20 minutes). Heat olive oil in a large nonstick skillet over medium-high heat. Cook falafel burgers until browned and heated through (3 minutes per side). Transfer to a plate. Before serving, whisk together tahini, lemon juice, and water in a medium bowl until light and fluffy. Divide falafel burgers, tahini mixture, lettuce, and roasted red peppers among pitas and serve.