Put the soup pot on the stove and turn the heat to medium. When the pot is hot, carefully add the oil.
Add the onion, carrots, and apple and cook, covered, until the vegetables are beginning to soften, about 15 minutes. Use the wooden spoon to stir them occasionally while they cook.
Turn up the heat to high, add the chicken or vegetable stock and bring to a boil. Reduce the heat to low, and cook until the carrots are completely tender, about 20 minutes. Set aside to cool for 20 minutes.
Use a slotted spoon to remove the solids, and put in the blender or food processor. Process until smooth. Add the yogurt, if you like, and process again, then stir this mixture back into the soup in the pot.
Serve right away or cover and refrigerate up to 3 days.