In a large pot, melt the butter over medium heat. Add the chopped onion and sauté for about 5 minutes, until translucent.
Add the minced garlic and thyme, cooking for an additional 1-2 minutes until fragrant.
Stir in the diced potatoes, and pour in the vegetable broth. Bring the mixture to a boil, then reduce heat to a simmer. Cook for 15-20 minutes until the potatoes are tender.
Once the potatoes are soft, use an immersion blender to puree the soup to your desired consistency (smooth or chunky). If you don't have an immersion blender, carefully transfer the mixture in batches to a regular blender.
Return the blended soup to the pot and stir in the heavy cream. Let the soup heat through for another 5-10 minutes on low heat.
Mix in the shredded cheddar cheese until melted and well-combined. Season with salt and pepper to taste.
Serve hot, garnished with fresh chives on top for a pop of color and flavor.
Notes
For a thicker soup, blend longer or add more potatoes.