Melt butter in a large stockpot medium heat. Add onion, and cook, stirring occasionally, until translucent, about 5 minutes.
Whisk in flour until lightly browned, about 1 minute.
Gradually whisk in milk and half-and-half, and cook, whisking constantly, until incorporated, about 1-2 minutes.
Stir in broccoli, nutmeg, salt and pepper, to taste.
Bring to a boil; reduce heat and simmer until broccoli is tender.
Carefully transfer broccoli mixture to blender, and blend smooth.
Return to stockpot.
Stir in cheese, a handful at a time, until smooth, about 1-2 minutes. Stir in vegetable broth until heated through.
Cook second broccoli bag in microwave, add to soup.
Serve immediately.