In a large pot or Dutch oven, heat olive oil over medium heat. Add the chopped onion and cook until softened, about 5 minutes. Stir in the minced garlic and cook for 1 more minute. Add the chicken breasts and cover with the chicken broth. Bring to a simmer, then reduce heat to low, cover, and cook for 20-25 minutes until the chicken is cooked through.

Remove the cooked chicken to a plate. Use two forks to shred it. Bring the broth back to a boil and stir in the orzo. Cook according to package directions, about 8-10 minutes, until al dente.

In a medium bowl, whisk the eggs until pale and frothy, about 1 minute. Continue to whisk while slowly streaming in the lemon juice. Temper the eggs by very slowly pouring in about 2 cups of hot broth from the pot, whisking constantly until combined.

Turn the heat of the soup pot to its lowest setting. Gently pour the tempered egg-lemon mixture back into the pot, stirring constantly. Add the shredded chicken. Continue to stir over low heat for 2-3 minutes to thicken, but do not allow it to boil. Stir in the fresh dill, season with salt and pepper, and serve immediately.