Add the cauliflower to a medium-size pot. Pour in chicken broth, cover, and bring to a boil. Turn down to a simmer and cook until very tender, about 10-12 minutes.
Add ½ of the cooked cauliflower to a blender or food processor. Add 1½ tablespoons each butter and sour cream. Add 1 tablespoon milk, ¼ teaspoon salt, and pepper, to taste. Blend until smooth. You’ll need to stop the blender a few times and push down the sides and blend again. Pour the first batch into a serving bowl or container.
Add remaining cauliflower, 1½ tablespoons of each butter, and sour cream to the blender. Sprinkle in ¼ teaspoon salt and pepper, to taste. Blend until very smooth, stopping to scrape down the sides to get all blended.
Add to a serving bowl and serve warm. This can be made a day before serving. Store covered in the refrigerator.