In a large pot, heat olive oil over medium heat. Add the chopped onion and sauté for about 5 minutes until translucent.
Stir in the minced garlic and cook for another minute until fragrant.
Add the diced potatoes to the pot and pour in the vegetable broth. Bring the mixture to a boil, then reduce the heat and simmer for 20-25 minutes, or until the potatoes are tender.
Once the potatoes are soft, use an immersion blender to puree the soup until smooth. If you prefer a chunkier texture, blend just half of the soup and leave the rest as is.
Stir in the heavy cream and shredded cheddar cheese. Heat through on low until the cheese is melted.
Season with salt, pepper, and smoked paprika (if using) to taste.
Serve the soup hot, topped with additional cheddar cheese, crispy shallots, and chopped green onions.
Notes
For a smoky flavor, add a pinch of smoked paprika.