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In a large pot or Dutch oven, melt butter over medium heat.
Add the onions and cook for 3 to 4 minutes, until they begin to soften.
Sprinkle the flour over the onions and stir to combine, cooking for about 1 minute.
Slowly whisk in the milk and half-and-half.
Add the nutmeg, broccoli, a dash of salt, and lots of freshly ground black pepper.
Cover and reduce the heat to low.
Simmer for 20 to 30 minutes, or until the broccoli is tender.
Add the cheese a handful at a time, stirring to melt the cheese after each addition.
At this point, blend the broccoli to your desired texture using an immersion blender, potato masher, or blender.
Finally, whisk in the vegetable broth, and re-warm over low heat, then serve.