Heat 1 tablespoon of vegetable oil in a large skillet over medium-high heat. Add the ground beef and cook until browned, breaking it up with a spoon as it cooks. Drain excess fat if needed and set the beef aside.
Add the remaining tablespoon of vegetable oil to the same skillet. Stir in the broccoli florets and cook for 3 to 4 minutes until they start to brighten and soften slightly. Add the minced garlic and ginger and stir for 30 seconds until fragrant.
Return the cooked ground beef to the skillet. In a small bowl, mix the soy sauce, brown sugar, cornstarch, and water until smooth. Pour the sauce over the beef and broccoli. Stir everything together and cook for 2 to 3 minutes until the sauce thickens and coats the ingredients evenly.
Remove the skillet from heat and drizzle with sesame oil. Serve the stir-fry over cooked rice.