In a large Dutch oven, melt the butter over medium heat.
Add the onion, carrots, ½ teaspoon salt, and pepper. Cook, stirring occasionally until softened, about 5 minutes. Add the garlic and cook, stirring often, for 2 minutes.
Add the flour and cook, stirring constantly, for 2 minutes.
Gradually whisk in the broth, breaking up any lumps. Stir in the half-and-half.
Increase the heat to medium-high and bring to a simmer, stirring occasionally. Once thickened, reduce the heat to low.
Stir in the broccoli, mustard powder, paprika, and remaining ½ teaspoon salt. Cover, and cook at a low simmer until the broccoli is tender, about 15 minutes.
Stir in the shredded cheddar a handful at a time until smooth. Taste, and add more salt, if desired. Serve immediately topped with additional cheddar, if desired.
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Notes
Make sure you add the shredded cheese in batches. Stirring between each handful of cheese added to the soup will help prevent clumping, and makes sure you evenly distribute the cheese throughout the soup.
Once the cheese has melted, avoid continuing to cook, as this can cause the cheese to separate or become grainy.
If you’d like a smoother soup, you can blend a portion of the soup.
Pre-shredded cheese can contain additives that might affect the texture of the soup. Grating the cheese yourself ensures you end up with a smoother and creamier consistency.
Make sure to chop the broccoli into small bite-sized florets, if they’re left too large, they won’t be tender from the cooking time.