In a medium stockpot, combine 6 cups chicken broth, 1/2 cup uncooked orzo or rice, and 1/2 teaspoon salt. Bring to a gentle boil over medium-high heat, then reduce heat to maintain a simmer. - If using orzo: simmer, stirring occasionally, until the orzo is tender, about 8–10 minutes. - If using rice: simmer, stirring occasionally, until the rice is tender, about 15–20 minutes or according to the rice package.
While the orzo or rice cooks, crack 3 large eggs (they should be at room temperature) into a medium bowl and whisk until smooth. Add 1/3 cup freshly squeezed lemon juice and whisk again until fully combined.
When the orzo or rice is cooked, remove the pot from the heat and let it stop actively boiling for 30 seconds.
Temper the egg-lemon mixture: using a ladle or heatproof cup, scoop one ladleful of hot broth from the pot and very slowly pour it into the egg-lemon mixture while whisking constantly. Repeat with a second ladleful, whisking continuously, until the egg mixture is warmed (this gradual warming prevents curdling).
Pour the tempered egg-lemon mixture back into the pot with the cooked orzo/rice and broth, whisking gently and continuously to combine. Do not return the pot to a boil.
Add 2 cups shredded chicken breast to the soup and warm gently over the lowest heat setting just until the chicken is heated through and the soup is hot enough to eat (1–3 minutes). Do not simmer or boil after adding the egg mixture to avoid curdling.
Taste and adjust seasoning only if needed.
Serve immediately, garnished with lemon slices, coarsely ground black pepper, and chopped fresh dill.