In a large pot, combine the butter and oil. Melt the butter, then add the oil, onion and garlic. Cook until the onions are soft, stirring occasionally.
Add the potatoes, cannellini beans, green onions, thyme and vegetable broth. Simmer for 25 minutes, or until potatoes are soft.
Using an immersion blender, blend the soup about 50%, leaving some chunks and texture.
If you don't have an immersion blender, remove about 1/2 the soup and puree it in a blender or food processor. Then return the pureed soup to the pot.
Add the half and half. Stir. Add salt and pepper to taste.
Serve and enjoy!