Cook onion, carrots, and celery: Heat olive oil in a Dutch oven or heavy-bottomed pot over medium. Add onion, carrots, celery, and ¼ teaspoon of the salt; cover and cook, stirring occasionally, until softened, 8 to 10 minutes.
Add chicken broth and rice: Stir in chicken broth and rice; bring mixture to a boil over medium-high. Reduce heat to medium; simmer, uncovered and undisturbed, until rice is almost tender, about 10 minutes.
Make egg mixture: Whisk together eggs and lemon juice in a bowl until frothy. Gradually stir 1 cup of hot broth from Dutch oven into egg mixture. Whisking vigorously, gradually add 1 cup additional hot broth to egg mixture, taking care not to add it too quickly to avoid scrambling the eggs.
Add egg mixture to soup: Whisk egg mixture into Dutch oven; stir in chicken and remaining ¼ teaspoon salt. Remove from heat, and stir until warmed through, about 2 minutes.
Garnish soup, and serve: Ladle soup into bowls. Top evenly with fresh dill and black pepper, and serve with lemon wedges, if desired.
