Melt butter in a large pot over medium-high heat. Stir in the onion, cooking for about 5 minutes.
Add garlic and carrot, cooking for 1 minute.
Stir in flour, mixing for a minute. This will give the soup a thicker consistency.
Add broccoli, paprika, salt, pepper, and chicken stock, stirring until everything is incorporated.
Bring mixture to a simmer, cover, and simmer for 10-15 minutes, until broccoli and carrots are tender.
Pour half of the soup into a blender. Place the top on the blender, but remove the center plastic cap and cover that area with a towel. This will prevent the hot liquid from forcing the lid off mid-blend.
Blend for about 30 seconds until the mixture is smooth. Add the blended portion back into the pot.
Add milk and cheese, stirring and cooking on medium heat for about 3-5 minutes or until warmed through.
Serve and enjoy!