In a large pot, heat the extra virgin olive oil over medium heat. Add the finely chopped onion, diced carrots, and diced celery. Sauté the mixture for approximately 5-7 minutes until the vegetables soften and the onion turns translucent.
Stir in the minced garlic and continue to cook for another minute until fragrant, being careful not to let it burn.
Pour the chicken or vegetable broth into the pot, increase the heat slightly, and bring to a gentle simmer. Allow it to cook for about 10 minutes, or until the vegetables are tender but still retain some bite.
Stir in the cooked rice, ensuring it is evenly distributed throughout the broth.
In a separate bowl, whisk the eggs until they become frothy and slightly pale in color. Gradually whisk in the fresh lemon juice and lemon zest, mixing continuously to achieve a smooth blend and prevent curdling.
To temper the eggs, gradually ladle out about 1 cup of the hot soup broth and slowly pour it into the egg mixture, whisking constantly as you do this. This step is crucial to avoid scrambling the eggs when you add them to the soup.
Slowly return the tempered egg mixture to the pot, stirring continuously for about 1-2 minutes. This will help the soup to thicken slightly and bring a luscious creaminess to the texture.
Season the soup generously with salt and freshly ground black pepper to taste, adjusting as desired. Remove from heat.
Serve the soup hot in bowls, and garnish each serving with a sprinkling of fresh dill or parsley for a burst of flavor and vibrant color.