In a large pot, melt butter over medium heat until bubbling. Add chopped onion and minced garlic, sautéing for 5-7 minutes until translucent.

Whisk in flour to the butter mixture, cooking for an additional 2-3 minutes until golden brown.

Gradually pour in broth while whisking to eliminate lumps. Add chopped broccoli florets and bring to a gentle simmer for 10-12 minutes.

Using an immersion blender, blend the soup to desired creaminess while leaving some broccoli pieces for texture.

Lower heat and stir in shredded cheddar cheese and milk or cream until melted and integrated.

Taste and adjust seasoning with salt and pepper, then garnish with fresh chives or parsley before serving.