Heat the vegetable oil and sesame oil together in a large skillet over medium-high heat until the oil shimmers.
Add the sliced beef and cook for 3 to 4 minutes, stirring often, until it is mostly browned but still slightly pink in spots.
Stir in the minced garlic and fresh ginger and cook for 30 seconds until fragrant.
Add the broccoli florets and pour in the soy sauce. Cook for 4 to 5 minutes, stirring frequently, until the broccoli turns bright green and stays crisp-tender.
Warm the cooked brown rice if needed, then divide it among four plates.
Spoon the beef and broccoli mixture over the rice and finish with the sliced green onions and sesame seeds.