In a small bowl, whisk together all of the ingredients to make the sauce.
Add broccoli to a vegetable steamer or saucepan with a few tablespoons of water. Cover and steam until tender. When done, remove from heat and let cool.
While broccoli steams - In a large skillet over medium heat, pour in the sauce and bring to a low simmer until sauce lightly thickens. When done, pour sauce back into the mixing bowl. Wipe skillet clean with a paper towel.
Thinly slice beef into ¼″ strips and toss in a large bowl. Toss beef with cornstarch. Add whisked egg to beef and toss to fully coat.
Heat oil in a large skillet over medium-high heat. Carefully add beef to the hot skillet and begin to stir fry the beef. Stir beef frequently until no longer pink. Remove cooked beef from skillet and set aside on plate.
Carefully wipe any excess grease from the skillet. Pour sauce into the skillet and heat up over medium-high heat. Toss beef and broccoli in sauce and stir fully coat. Remove from heat.
Serve Mongolian Beef and Broccoli with fresh chopped green onions and sesame seeds(optional) over steamed rice.