In a large Dutch oven, melt butter over medium heat. Add diced onion and sauté for 3 minutes until translucent.

Sprinkle in flour, stirring continuously for about 2 minutes until golden and bubbly. Incorporate minced garlic and cook for an additional 1-2 minutes.

Pour in chicken broth and milk, stirring well. Add chopped broccoli, paprika, ground mustard, salt, and pepper.

Bring the mixture to a gentle simmer over medium-high heat, then reduce to low, cover, and simmer for 20 minutes.

Blend half of the soup using an immersion blender until smooth. Return the mixture to the pot.

Gradually stir in shredded cheddar cheese until melted. Adjust consistency with more milk if needed.

Taste and adjust seasoning with additional salt and pepper if needed before serving.