In a large pot, melt the butter over medium heat. Add the chopped onion, celery, and carrot. Sauté for about 5 minutes until softened.

Add the minced garlic and cook for an additional 1 minute until fragrant.

Add the diced potatoes, vegetable broth, thyme, parsley, salt, and pepper to the pot. Bring to a boil, then reduce heat and simmer for 20 minutes, or until the potatoes are tender.

Using an immersion blender, blend the soup until smooth. Alternatively, carefully transfer the soup in batches to a blender and blend until smooth.

Stir in the milk and shredded cheddar cheese until the cheese is melted and the soup is creamy. Adjust seasoning if necessary.

Ladle the soup into bowls and garnish with additional cheese or herbs if desired. Serve hot.