Bring the broth to a boil in a medium saucepan. Add rice; cook uncovered, 10 minutes. Drain rice in a colander over a bowl, reserving 1/4 cup broth; set the rice and broth aside.
While the rice is cooking, heat the sesame oil in a large skillet over medium-high heat. Add the green onions, vinegar, garlic, and ginger; saute for 1 minute. Add mushrooms, sauté 2 minutes. Stir in rice, reserved broth, and chicken. Cook 3 minutes or until the liquid almost evaporates, stirring frequently.