Place a large nonstick skillet over medium-high heat until hot. Add 1 teaspoon sesame oil to pan, and tilt to coat evenly. Add turkey, and stir-fry 5 minutes or until turkey is no longer pink in center. Remove turkey, and set aside.
Combine broth and next 4 ingredients in a small bowl; stir until cornstarch dissolves. Set aside.
Add remaining 1 teaspoon oil to pan; add pepper strips and broccoli; stir-fry 1 minute. Add water chestnuts, and stir-fry 30 seconds. Increase heat to high. Stir broth mixture, and add to pan with soy sauce, turkey, and any accumulated juices. Bring to a boil; cook 1 to 2 minutes or until slightly thickened.