Heat the olive oil in a 5.5-quart / 5.2 L Dutch oven over medium heat. Add the onion, carrots, and celery, and cook for 5 to 7 minutes, stirring often, until softened.

Add the garlic and cook for 30 seconds, until fragrant.

Pour in the chicken broth and add the chicken breasts. Bring to a boil, then lower the heat and simmer, covered, for 15 to 18 minutes, until the chicken reaches 165 F / 74 C on an instant-read thermometer.

Lift the chicken onto a plate. Once it's cool enough to handle, shred it into bite-sized pieces with two forks and set aside.

Stir the rice into the pot and simmer uncovered for 15 to 18 minutes, until the rice is tender.

While the rice cooks, whisk the eggs and lemon juice together in a medium bowl until pale and frothy.

Ladle about 1 cup of hot broth from the pot into the egg mixture in a slow stream, whisking the whole time, to warm the eggs without cooking them. Repeat with a second cup of broth.

Turn the heat down to low. Slowly pour the tempered egg mixture back into the pot, stirring constantly.

Return the shredded chicken to the pot. Warm everything gently for 3 to 4 minutes, stirring, until the soup thickens slightly and reaches about 160 F / 71 C. Do not let it boil, or the eggs will curdle into small scrambled bits instead of thickening the broth.

Season with salt and pepper, taste, and adjust. Ladle into bowls and top with chopped dill.