Sauté the aromatics: In a large pot, melt the butter over medium heat. Add the chopped onion and cook for about 3 to 4 minutes until softened and lightly golden. Add the minced garlic and sauté for another minute until fragrant.
Make the roux: Sprinkle in the flour and whisk for 1 to 2 minutes until the mixture turns golden. This step helps thicken the soup and gives it that creamy texture.
Add the broth and seasonings: Pour in the chicken or vegetable broth, then stir in the salt, pepper, and paprika (or nutmeg). Bring the mixture to a boil.
Simmer the veggies: Add the chopped broccoli and grated carrot to the pot. Reduce the heat to medium-low and let it simmer for about 15 minutes, or until the broccoli is tender.
Finish with cream and cheese: Stir in the half & half and freshly grated cheddar cheese. Continue to cook for another minute until the cheese is melted and the soup is smooth. Taste and adjust seasoning if needed.