1. Cut scallions in half crosswise, dividing white and green parts. Coarsely chop white parts and set aside. Finely chop green parts and set aside separately.
2. Heat oil in a medium pot over medium heat. Add onions and chopped white parts of scallions and cook, stirring often with a wooden spoon, until soft, 8-10 minutes. Add potatoes and stock and season to taste with salt and pepper. Increase heat to medium-high and bring just to a boil. Reduce heat to medium-low ans simmer, stirring occasionally, until potatoes are soft, 30-35 minutes.
3. Allow vegetables and stock to cool slightly; then, working in batches, puree them together in the jar of a blender or the work bowl of a food processor until very smooth. Return puree to pot and cook over medium heat until hot. Adjust seasonings. Garnish soup with reserved scallion greens.