Broccoli & Beef is believed to originate from a Chinese dish called "Gai Lan Chao Niu Rou" (芥兰炒牛肉)- also known as Chinese Broccoli Fried Beef. Early Chinese immigrants could not find Gai Lan (Chinese variant of broccoli) in the US and so the broccoli that we know today became an easy substitute.
It is believed that the first versions of the dish were prepared by Chinese immigrants who had settled in San Francisco sometime after the 1920's when broccoli was introduced by immigrants from Messina.
Fun fact: Broccoli became popular in the US around that time as well! Over time, this recipe became popular all around the world for its nutrition and tender beef texture.
Here's my dad's recipe where you can make Chinese restaurant-level Broccoli & Beef in your own kitchen!