Add the chopped onion and sauté until it becomes translucent, about 5 minutes.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the diced potatoes to the pot, pouring in the vegetable broth. Bring to a boil, then reduce to a simmer.
Cook the potatoes covered for about 15-20 minutes or until they are fork-tender.
Remove the pot from heat and, using an immersion blender, blend the soup until creamy. (Alternatively, transfer the mixture to a blender in batches but be cautious of hot steam.)
Return the pot to low heat and stir in the heavy cream, grated cheddar cheese, thyme, salt, and pepper. Stir until the cheese is melted and the soup is heated through.
Taste and adjust seasoning if necessary.
Serve hot, garnished with chopped green onions or chives.