Remove the green parts from the leeks, slicing only the white sections into thick rounds.
In a deep, large pan or wide pot over medium heat, add the leek slices along with a quarter cup of water. Steam until they soften and absorb most of the water.
Add olive oil and the chopped onion to the pan, keeping the heat medium. Sauté until the leeks and onion are translucent, releasing their natural sweetness.
Finely chop the white part of the spring onions, add to the pan, and sauté for another 1-2 minutes.
Add the rice, stirring for about a minute to coat it with oil.
Pour in the warm water and, once it begins to boil, add the grated carrot. Lower the heat to a gentle simmer.
Partially cover the pot and simmer for about 17 minutes, stirring occasionally to avoid breaking up the leeks. Season with salt and pepper towards the end of cooking.
At around 15-16 minutes, add 2-3 tbsp olive oil, chopped dill, green parts of the spring onions, and the lemon juice. Stir well and remove from heat.
Cover the pot with a clean towel, place the lid on top, and let it sit for 10 minutes to allow the rice to absorb the remaining moisture.
Pair the leek rice with a bit of Greek yogurt on the side. Optionally, you can make a variation by adding chopped or grated tomatoes to the same ingredients for a tomato-based version.