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.
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 and simmer, stirring occasionally, until potatoes are soft, 30–35 minutes.
Allow vegetables and stock to cool slightly; then, working in batches, purée them together in the jar of a blender or the work bowl of a food processor until very smooth. Return purée to pot and cook over medium heat until hot. Adjust seasonings. Garnish soup with reserved scallion greens.
Notes
This is the perfect soup for making ahead to eat all week. You can also freeze individual servings for quick lunches.