Melt the butter in a Dutch oven or soup pot over low heat. Add the onion and the garlic and stir until they glisten with butter, then season with salt and 1 teaspoon white pepper, cover, and cook for about 10 minutes, until the onion is soft but not colored.
Add the remaining ingredients, along with a little more salt, increase the heat, and bring to a boil. As soon as the soup bubbles, turn the heat to low, mostly cover the pot, and simmer gently for 30-40 minutes, or until all the vegetables are mashably soft. Taste the soup and season generously with salt and white pepper.
The soup can be pureed at this point in a blender or with an immersion blender, or can be served chunky (we pureed half and added it back in so the soup thickened, and there were still chunks of potato).