Melt butter in a large pot over medium heat.
Add chopped onion and sauté until soft and translucent.
Stir in minced garlic and cook for 30 seconds.
Sprinkle flour over the mixture and whisk continuously.
Slowly pour in vegetable broth while whisking to avoid lumps.
Add broccoli florets and bring the soup to a gentle simmer.
Cook for 12–15 minutes until broccoli is tender.
Reduce heat and stir in milk.
Add shredded cheddar cheese gradually, stirring until melted.
Season with salt, black pepper, and nutmeg.
Remove from heat and serve warm.