In a large pot, heat the olive oil or butter over medium heat. Add the diced onion and carrot; cook until the onion is translucent and the carrot softens, about 5 minutes.

Add minced garlic and cook for 30 seconds until fragrant, then pour in the chicken broth and bring to a gentle simmer.

Add rice or orzo to the simmering broth. Reduce heat to maintain a gentle simmer and cook until the rice is tender but still slightly firm, about 15–20 minutes.

Stir in the shredded chicken and warm through for 2–3 minutes, then taste and add salt and pepper as needed.

In a medium bowl, whisk the eggs until frothy. Slowly whisk in the lemon juice until fully combined.

Ladle about a cup of hot broth into the egg-lemon mixture, whisking constantly to prevent curdling. Continue adding hot broth slowly until the bowl feels warm and the mixture has tempered.

Slowly pour the tempered egg-lemon mixture back into the pot while stirring the soup gently. Heat on very low for 1–2 minutes—do not boil—or the eggs may curdle.

Stir in a bit of lemon zest and additional lemon juice for more tanginess, along with a generous handful of chopped dill or parsley.

Ladle into bowls and top with extra black pepper and a drizzle of olive oil if desired.