Melt 1 tablespoon butter in a large soup pot or Dutch oven over medium heat. Add the minced onion, garlic, and chopped broccoli stems, then cook until the vegetables are softened and fragrant, about 5 minutes.

Add the remaining 4 tablespoons butter to the pot. Once melted, stir in the flour and cook over medium heat for 2-3 minutes until the mixture thickens.

Gradually whisk in the milk and half and half, then slowly add the broth while continuing to whisk. Simmer for about 10 minutes, whisking occasionally to keep the soup base smooth.

Stir the broccoli florets, carrots, salt, pepper, and paprika into the thickened soup base. Simmer for about 10 minutes, until the broccoli is bright green and fork-tender.

Remove the pot from the heat and let the soup cool slightly for a few minutes. Stir in most of the grated cheddar until melted, then serve with a little extra cheese if desired.