Combine beef, garlic, ginger and soy sauce in a bowl and let stand.
Wash broccoli thoroughly and cut into florets. Trim and peel stems and cut into 1/4-inch thick slices.
Heat 1 Tbsp. oil in a large nonstick skillet or wok over high heat, add broccoli florets and stems, then stir-fry for 2 minutes. Add 1/2 cup water and stir until water evaporates. Transfer broccoli to a plate.
Add remaining oil to pan, add beef mixture and stir-fry for 3 minutes. Stir together broth and cornstarch, add to meat and stir-fry until sauce is thickened, about 3 minutes longer. Add broccoli and bean sprouts, then cook, stirring, until heated through, about 2 minutes.