Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add broccoli and 2 teaspoons ginger to pan; sauté 1 minute. Add water. Cover; cook 2 minutes or until broccoli is crisp-tender. Remove broccoli from pan; keep warm.
Heat remaining 2 teaspoons oil in pan; add remaining 1 teaspoon ginger, crushed red pepper, and chicken. Cook 4 minutes or until chicken is lightly browned, stirring frequently.
Combine broth and next 5 ingredients (through garlic) in a small bowl, and stir with a whisk. Add broth mixture to pan; cook 1 minute or until mixture thickens, stirring constantly. Return broccoli mixture to pan; toss to coat. Sprinkle with peanuts.