Chop beef and broccoli. Set broccoli aside and place beef into slow cooker. Add broth, soy sauce, minced garlic, brown sugar, sesame oil, and chili flakes to slow cooker. In a small bowl whisk corn starch into water until dissolved. Add to slow cooker and stir. Cover and cook on high for 2-3 hours or low 4-5 hours.
Just before beef is finished, steam broccoli by adding 1/2 cup of water and the chopped broccoli to a small pot. Cover and bring to a boil. Cook on high for 2-3 minutes. Remove from heat, drain, and stir broccoli into the slow cooker.