Heat butter in a large pot over medium heat.
Add chopped onions and garlic; sauté until soft.
Stir in the diced potatoes and cook for a few minutes.
Pour in the vegetable broth and bring to a boil.
Reduce heat and let simmer for about 20 minutes or until the potatoes are tender.
Using a potato masher, mash some of the potatoes for a creamy texture.
Stir in milk and shredded cheddar cheese until melted.
Add salt and pepper to taste.
Simmer for an additional 5 minutes to thicken.
Serve hot, garnished with chopped chives and croutons.