01 - Melt the butter in a large pot over medium heat. Add the diced onion, carrots, and celery. Sauté for 5 minutes until softened and fragrant.
02 - Stir in the minced garlic and cook for 1 minute until fragrant, being careful not to burn it.
03 - Sprinkle the flour over the vegetables and cook, stirring constantly, for 2 minutes to form a roux and eliminate the raw flour taste.
04 - Gradually pour in the vegetable broth while whisking continuously to prevent lumps from forming.
05 - Add the cauliflower florets and bring the mixture to a boil. Reduce heat to low and simmer for 15 minutes until the cauliflower is very tender.
06 - Use an immersion blender to partially blend the soup, leaving some texture for a hearty consistency. Alternatively, transfer half the soup to a blender, blend until smooth, and return to the pot.
07 - Stir in the whole milk and heavy cream. Simmer for 3 minutes to heat through and blend flavors.
08 - Remove from heat and stir in the shredded cheddar cheese until melted and smooth. Season with salt, black pepper, and smoked paprika to taste.
09 - Ladle the chowder into bowls and garnish with fresh chives and extra shredded cheddar if desired. Serve hot with crusty bread.