In a large stockpot, heat the olive oil over medium heat, and add the shallots. Cook, stirring frequently, until shallots are soft and translucent, about 5 minutes.
Add cauliflower and potatoes to stockpot. Cover with cold water. Turn heat to high, and bring to boil. Reduce heat and cook at a simmer, uncovered, until the cauliflower and potatoes are tender, 15 minutes.
Remove pot from heat and drain, reserving a few cups of the hot cooking liquid. Add butter, broth, salt, and pepper. Working in batches, puree soup in a blender until smooth. Add additional liquid as needed to reach desired consistency (fairly thin is recommended). To serve, ladle into warm bowls and drizzle each por-tion with a teaspoon of extra-virgin olive oil, or more to taste.