Cut broccoli florets and stems into bite-size pieces to measure 10 cups.
Bring water to a boil in a large Dutch oven; add broccoli. Cook broccoli 4 minutes or until crisp-tender; drain.
Combine broth and next 6 ingredients (broth through sesame oil); stir well with a whisk. Heat the vegetable oil in a large nonstick skillet over medium-high heat. Add onions, ginger, and garlic; sauté 15 seconds. Add broth mixture; bring to a boil. Cook 1 minute or until thick, stirring constantly. Add broccoli; cook 30 seconds, tossing to coat.