In a Dutch oven, melt 1 tablespoon of butter over medium heat. Add chopped onion and sauté until soft and translucent, about 2-3 minutes.

Add 1/4 cup of butter to the Dutch oven and melt it over medium heat. Once melted, add flour and whisk together. Cook the mixture for about 1 minute, whisking often.

Slowly pour in about a cup of chicken broth, whisking as you pour. Continue with the remaining chicken broth and half and half, pouring slowly and whisking constantly.

Let the broth/half and half mixture cook for a minute or two, whisking often to ensure no flour lumps remain.

Add broccoli and carrots to the pot and stir to combine. Add paprika, garlic powder, dry mustard, salt, and pepper, and stir again. Reduce the heat to low/medium-low and simmer for about 10-15 minutes or until thickened and the broccoli and carrots are fork-tender. If the soup becomes too thick, you can add a splash of chicken broth or half and half to loosen it up.

Blend the soup until you reach your desired texture, using either an immersion blender or by transferring the soup to a traditional blender.

Gradually add the grated cheddar cheese to the soup, stirring to melt the cheese into the soup after each addition.

Taste the soup and season with additional salt and pepper if needed.

Serve hot, garnished with additional black pepper and shredded cheese if desired. Enjoy!