In a stock pot over medium heat, melt butter; add onion; cook onion ~4 minutes; sprinkle in flour, stirring to combine; cook 1 minute more.
Add broccoli, chicken broth, garlic, thyme, nutmeg, salt and pepper; bring to a bubbly boil; cover and reduce the heat to low; simmer ~30 minutes.
Stir in cheese; allow to melt; pour in milk and half-andhalf; heat through on medium-low heat.
If you wish to break down the broccoli, proceed to step 5-6.
Transfer all to a blender in batches and purée.
Return all to stock pot; heat through on medium-low heat.