Slice thick broccoli stems into thin coins. Cut thin stems and florets into 1" - 2" pieces. Heat the oil in a large skillet over medium high heat. When it begins to shimmer add the stems and cook for 3 to 4 minutes until they are beginning to brown in places but are still bright green. Add the remaining broccoli pieces and stir-fry until they are bright green and just beginning to soften but remain crisp. Remove to a plate.
Crumble the ground beef into the skillet and break into pieces with a wooden spoon. Cook over medium high heat until no pink remains. Use a microplane or other grater and grate the ginger and garlic into the meat. Add ¾ of the slices green onions and stir. Cook, stirring often, until the beef is well browned, about 10 minutes total for this step. Reduce the heat to medium and add the cooked broccoli back into the skillet.
In a small bowl, stir together the soy sauce, broth, sugar, and cornstarch until the sugar is completely dissolved. Add the soy mixture to the meat and broccoli, stirring until the sauce slightly thickens.
Serve over rice, and garnish with the remaining green onion slices.