In a large pan, steam the broccoli until the stems can be pierced with a fork. They should be firm, not too soft.
In a large pan filled halfway with water, boil the potatoes and onions until the potatoes can be easily pierced with a fork. Drain the water and add the broccoli to the potatoes and onions.
Pour the vegetable broth into the pan with the potatoes and broccoli and lower the heat to medium low. Use an immersion blender to blend the soup to a creamy texture. (If you don’t have an immersion blender you can use a standing blender or a food processor, just be sure to let it cool a bit before blending.)