In a large pot, heat the olive oil over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the diced potato and grated carrot to the pot, stirring to combine. Cook for about 5 minutes.
Pour in the vegetable broth and water, bringing the mixture to a boil. Reduce heat and let it simmer for 15 minutes, or until the potatoes are tender.
Add the chopped kale or spinach and fresh dill to the pot. Simmer for another 5 minutes until the greens are wilted.
Stir in the lemon juice, and season with salt and pepper to taste.
Serve hot, garnished with a dollop of sour cream and sliced hard-boiled eggs if desired.