Cook the potatoes and onions (tops and all) in chicken broth until tender. Puree in blender, a little at a time, while it's hot. A hand blender works best because the whole mixture can be pureed right in the pan. Steam broccoli until it's tender. Dice it and add with salt, thyme, dry mustard, basil and nutmeg to the pureed mixture. Gradually stir in Half and Half. When ready to serve, sprinkle grated cheese on top. Sometimes I stir a little cheese into the soup. This can be made a day ahead and reheated.