In a large pot, heat the olive oil over medium heat. Add the chopped onion and garlic, sautéing for about 3-4 minutes until fragrant and soft.
Add the diced chicken to the pot, cooking until it's browned and cooked through, approximately 5-7 minutes.
Stir in the diced carrots and cook for an additional 3 minutes.
Pour in the chicken broth and bring the mixture to a boil. Once boiling, add the orzo and reduce the heat to a simmer. Cook for about 10 minutes or until the orzo is al dente.
In a separate bowl, whisk together the eggs and lemon juice until well combined. Gradually add a ladle of the hot soup broth into the egg mixture while whisking continuously to temper the eggs. This prevents them from scrambling.
Slowly stir the tempered egg mixture back into the pot of soup, continually stirring. This will create a creamy, lemony texture.
Stir in the fresh dill, and season with salt and pepper to taste. Allow the soup to heat through for another 2-3 minutes without boiling.
Serve hot, garnished with fresh parsley for added color and flavor.