Whisk together the soy sauce, sesame oil, honey, garlic, and ginger.
Place the beef in a zipped plastic bag (or glass bowl), pouring in half of the sauce mixture to coat the beef. Marinate for either 20 minutes on the counter or covered overnight in the fridge.
Whisk the cornstarch and water together and add to the remaining sauce mixture.
Heat a wok or large sauté pan over high heat, add 2 teaspoons oil and add the beef. Cook for 4 minutes and remove beef to a clean bowl.
Add the remaining 1 teaspoon of oil to the pan and cook the broccoli for 2 minutes.
Add the beef back into the pan. Make a well in the center and pour in the sauce. Cook until the sauce is thick and everything is heated through, about 2 minutes.
Serve over brown rice.