To make sauce, whisk together soy sauce, water, brown sugar, ginger, garlic, cornstarch, sesame oil, and chile-garlic sauce in a small bowl; set aside. Heat 2 teaspoons vegetable oil in a large skillet over medium-high heat. Add steak and cook until browned and just cooked through (2-3 minutes per side). Transfer steak (with its juices) to a bowl. Add 2 teaspoons vegetable oil to skillet with whites from scallions; cook, stirring, until charred (2 minutes). Add shiitake mushrooms and cook until softened (3-4 minutes). Add snow peas, scallion greens, and reserved sauce to skillet with reserved beef mixture; cook until heated through and sauce thickens (2-3 minutes). To assemble, arrange whole-wheat wraps on work surface; place about 3/4 cup beef mixture in middle of each and wrap tightly.