In a large pot, melt 1 tablespoon of butter over medium heat. Add chopped carrots, celery, and diced onion. Sauté for about 4 minutes until tender, then stir in minced garlic and cook for an additional minute.

Pour in 4 cups of low-sodium chicken broth, followed by cubed potatoes, celery root, thyme, salt, pepper, paprika, and cayenne. Bring to a boil, then reduce to a simmer and cover, cooking for 15 minutes.

Stir in chopped broccoli and cauliflower florets, letting the mixture simmer for an additional 5-7 minutes until the vegetables are tender.

In a separate saucepan, melt 4 tablespoons of butter. Whisk in 6 tablespoons of flour to create a roux, then gradually whisk in 3 cups of milk until thick and smooth.

Remove the creamy mixture from heat and slowly stir it into the vegetable mixture until well combined.

Fold in 2 cups of shredded cheddar cheese until melted and the chowder is silky. Adjust seasoning if desired.

Serve the chowder warm in bowls, best enjoyed immediately with crusty bread.