Chop the onion and carrots into small, even pieces. Cut the broccoli into bite-sized florets, keeping a bit of stem on each piece for texture. Mince or press the garlic and set it aside near the stove.

Melt the butter in a large Dutch oven over medium heat. Add the chopped onion and carrot, plus half a teaspoon each of salt and black pepper. Cook for about 5 minutes, stirring occasionally, until the vegetables are tender.

Stir the minced garlic into the pot and cook for 2 minutes, stirring constantly, until fragrant.

Sprinkle the flour into the pot and stir well to coat the vegetables. Cook for 2 minutes, stirring constantly, until the flour loses its raw taste and the mixture becomes golden and thick.

Slowly pour in the chicken broth while whisking to keep the mixture smooth. Once combined, whisk in the half-and-half slowly and steadily.

Turn the heat up to medium-high and let the soup come to a gentle simmer, stirring occasionally. The broth will start to thicken slightly, but it will still be fairly thin.

Stir in the chopped broccoli, paprika, mustard powder, and the remaining half teaspoon of salt. Mix well, cover the pot, and reduce the heat to low. Let the soup simmer for about 15 minutes, until the broccoli is tender.

Lower the heat, then add the shredded cheddar a handful at a time, stirring after each addition until fully melted before adding more.

Taste the soup and add extra salt or pepper if needed. Ladle the soup into bowls and top with a small handful of extra shredded cheese.