Pull the dill fronds from the stems; reserve 1/2 cup of the fronds for garnish.
In a pot, combine the stems, chicken thighs and rice with the chicken broth and bring to a boil over high heat.
Lower the heat to medium-low and stir. Cover and cook the chicken until tender, 25 minutes.
Use a slotted spoon to remove the chicken and stems from the broth. Discard the stems and transfer the chicken to a plate then shred with 2 forks.
Meanwhile, increase the heat to medium-high and bring the stock up to an active simmer.
In a large heatproof bowl, vigorously whisk the eggs until frothy. Add ¼ cup lemon juice and mix to incorporate.
While you whisk the eggs, gradually drizzle in 1 ladleful (about 1 cup) of the broth (it’s OK if some grains of rice tag along) and mix until fully combined.
Pour the lemon-egg mixture into the pot with the broth, immediately remove from the heat, and stir in the shredded chicken. Taste and add more lemon juice to taste.
Garnish with a generous amount of fresh dill, black pepper and a drizzle of olive oil.
Notes
If you wish to use chicken breast, that would be fine too.