Heat oil in a small pot over medium heat. Add onion and cook, stirring occasionally, until soft and light golden, about 5 minutes. Stir in garlic and ginger and cook, stirring constantly, for 1 minute. Add broth, potato, lentils, and carrots; bring to a simmer and cook, partially covered and stirring occasionally, until lentils and vegetables are tender, about 15 minutes. Stir in spinach and season to taste with salt and pepper.
Meanwhile, stir yogurt and chives in a small bowl; season to taste with salt and pepper. Ladle soup into bowls and serve with a dollop of chive yogurt.