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In a Dutch oven over medium heat, melt the butter. Add the chopped yellow onion and cook, stirring often, until softened, about 3 minutes.
Add the minced garlic and cook for 1 more minute until fragrant.
Whisk in the flour and cook, stirring constantly, until the mixture turns a light tan color, about 5-7 minutes.
Gradually pour in the chicken broth, a little at a time, whisking continuously and allowing the flour to absorb the liquid between additions.
Once all the broth has been incorporated, pour in the half and half cream and stir until smooth.
Bring the soup to a simmer. Season with salt, ground mustard, smoked paprika (if using), and a pinch of black pepper.
Once the soup is bubbling gently, reduce the heat and let it simmer for 5 minutes.
Add the broccoli and shredded carrots. Bring back to a simmer and cook uncovered, stirring occasionally, for about 20 minutes or until the broccoli is tender.
Remove the pot from heat and stir in the shredded cheddar cheese until melted and well combined.
Serve immediately, topped with extra cheddar cheese if desired.
Serving and Storage Tips:
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