In a large pot, melt the butter over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.
Pour in the vegetable broth and add the diced potato. Bring to a boil, then reduce heat and simmer until the potatoes are tender, about 15 minutes.
Add the chopped butter lettuce to the pot and cook for another 5 minutes until wilted.
Remove the pot from heat and use an immersion blender to puree the soup until smooth. Alternatively, carefully transfer the soup in batches to a blender and blend until smooth.
Return the soup to low heat and stir in the heavy cream. Season with salt and pepper to taste. Heat through but do not boil.
Serve hot, garnished with chopped chives.