Melt butter over medium heat in a large pot. Sauté diced onion until translucent, about 5 minutes. Add minced garlic and cook for an additional minute.

Incorporate diced potatoes and pour in broth. Bring to a boil, then reduce heat to medium-low and simmer for 15-20 minutes until potatoes are fork-tender.

Partially mash the potatoes using a potato masher or blend with an immersion blender to achieve desired thickness.

Stir in heavy cream and shredded cheddar cheese until melted and creamy. Season with thyme, salt, and pepper.

Ladle soup into warm bowls; garnish with fresh herbs or extra cheddar cheese if desired. Serve hot.