Start by washing the broccoli. Cut it into bite-sized florets. Chop your onion and celery finely. Grate the cheddar cheese if you are not using pre-shredded cheese. Prepare all your ingredients before you start cooking to make the process smoother.

In a large pot, melt the butter over medium heat. Add the chopped onion and celery. Sauté for about 5 minutes until the onion becomes translucent. This step is crucial. It builds the soup's base flavor. Stir occasionally to prevent burning.

Sprinkle the flour over the sautéed vegetables. Stir well so the flour coats everything. Cook for about 2 minutes. This allows the flour to lose its raw taste and start forming a roux, which will help thicken the soup later.

Gradually pour in the chicken broth. Use a whisk to blend everything, making sure there are no lumps. Increase the heat and bring this mixture to a gentle boil. Reduce the heat once it bubbles, allowing it to simmer for about 5 minutes.

Toss in the broccoli florets. Stir to combine. Let the soup simmer for another 10-15 minutes until the broccoli is tender but still bright green. This step is about bringing out the freshest flavors while keeping that fantastic color.

Once the broccoli is tender, lower the heat. Gradually stir in the whole milk. Add the Dijon mustard, smoked paprika, salt, and pepper. Let this warm through for about 5 minutes. Do not let it boil again, as this can cause the milk to curdle.

For a creamier texture, you can blend about half of the soup using an immersion blender or a regular blender. If you prefer a chunkier soup, skip this step.

Finally, remove the pot from the heat. Stir in the shredded cheddar cheese until it melts and combines fully. Taste and adjust seasoning if needed.

Ladle the soup into bowls. Top with additional shredded cheddar for that cheesy finish. You can add a sprinkle of smoked paprika or fresh cracked pepper for visual appeal.