Wash the broccoli carefully under cold water. Chop the florets into small bite-sized pieces. Grate the carrot and finely chop the onion. Smaller pieces cook faster and create a better soup texture. Preparing everything first makes the cooking process smooth and stress-free.

Place a large soup pot on medium heat. Add butter and let it melt slowly. Add the chopped onion and cook for about three minutes. Stir gently until the onion becomes soft and fragrant. This step builds the base flavor for the soup.

Add flour to the melted butter and onion mixture. Stir constantly using a whisk or wooden spoon. Cook for about two minutes. The mixture will become thick and slightly golden. This simple roux helps thicken the soup later.

Slowly pour the chicken broth into the pot while stirring. Then add the milk and cream. Continue stirring so the mixture stays smooth. The soup base will begin to look creamy and silky as it warms.

Add chopped broccoli and grated carrot to the pot. Stir gently so the vegetables mix evenly. Let the soup simmer for about ten minutes. The broccoli will soften and release its fresh flavor into the creamy broth.

Reduce the heat to low before adding the cheese. Slowly sprinkle the shredded cheddar cheese while stirring constantly. This helps the cheese melt smoothly without clumping. The soup will turn rich, thick, and beautifully cheesy.

Add garlic powder, salt, black pepper and paprika. Stir well to blend the flavors. Taste the soup and adjust seasoning if needed. If you want extra creaminess, add cream cheese and stir until melted.

For a smoother soup, blend part of the soup using an immersion blender. Some people prefer small broccoli pieces for texture. Choose the style you like best. Both options taste delicious and comforting.

Let the soup simmer for five more minutes so the flavors blend together. Stir occasionally to prevent sticking. Once thick and creamy, serve the soup hot in bowls with bread or crackers.