Heat oil in a medium stock pot. Add celery, carrots, onion and garlic. Cook over medium heat, stirring often until vegetables are soft, about 5 minutes.
Add broth, potato and lentils. Cover and simmer 30 to 40 minutes, stirring occasionally until potatoes and lentils are tender.
With a spoon or potato masher, press potatoes and lentils to make a slightly thickened soup. Stir in kale and cook 5-10 minutes until tender. Top with walnuts just before serving.