Melt the butter in a large Dutch oven or heavy-bottomed pot over medium heat. Add the diced onion and cook, stirring occasionally, for 5 to 6 minutes until softened and slightly translucent.

Add the minced garlic and cook for 1 minute more, stirring constantly, until fragrant.

Sprinkle the flour over the onion and garlic mixture. Stir and cook for 2 full minutes, stirring constantly, to cook out the raw flour taste and form a smooth roux.

Slowly pour in the chicken broth a little at a time, whisking constantly to prevent lumps from forming. Once all the broth is incorporated and smooth, pour in the whole milk and heavy cream, whisking to combine.

Bring the mixture to a gentle simmer over medium heat, stirring frequently. The soup will begin to thicken after about 3 to 4 minutes.

Add the broccoli florets and shredded carrot to the pot. Stir to combine, then reduce heat to medium-low. Cover partially and simmer for 15 to 20 minutes, or until the broccoli is completely tender and easily pierced with a fork.

If you prefer a smoother soup, use an immersion blender to partially blend the soup directly in the pot, pulsing 3 to 4 times to break down some of the broccoli while leaving chunks. For a chunkier texture, skip this step.

Remove the pot from the heat. Add the Dijon mustard, smoked paprika, ground nutmeg, hot sauce, salt, and pepper. Stir to combine.

Add the freshly shredded cheddar cheese in three additions, stirring well after each addition until fully melted and smooth before adding the next. The residual heat from the soup will melt the cheese perfectly without it seizing up.

Taste and adjust seasoning with additional salt, pepper, or hot sauce as needed.

Ladle the soup into bowls and top with extra shredded cheddar, a dollop of sour cream, or croutons if desired. Serve immediately with crusty bread or rolls for dipping.