Melt butter in a large pot over medium heat. Add onion and cook, stirring occasionally, until soft and translucent, 4-5 minutes. Add garlic and cook 1 minute more.
Add cauliflower and potato and sauté for 2 minutes, stirring occasionally. Add vegetable stock, butter beans, and leaves from 2 sprigs of thyme; stir. (Tip: Before adding stock, remove and set aside a few small florets of sautéed cauliflower for garnish later.)
Bring to a boil, then reduce heat and simmer for 20 minutes.
Carefully pour the soup into a blender or food processor and blend until smooth. (Alternately, puree with an immersion blender until creamy and smooth).
Stir in heavy cream and check for seasoning. Season with salt and pepper, to taste. Adjust consistency as needed (e.g., add more vegetable stock to thin out).
Garnish with coarsely ground black pepper, fresh thyme, and a drizzle of olive oil and/or heavy cream. Serve hot with warmed Martin’s Dinner Potato Rolls and salted butter.