Add to the crockpot: chicken, broth, onion, carrots, celery, garlic, rice, lemon zest, bay leaves, oregano, salt, and pepper. Stir to distribute the rice.
Cook on Low for 6–7 hours or High for 3–4 hours, until the chicken is very tender and the rice is soft.
Remove the chicken and bay leaves.
Shred the chicken with two forks, then set aside.
Make the lemon-egg mixture: In a medium bowl, whisk the eggs until frothy, then slowly whisk in the lemon juice. This is your avgolemono base.
Temper the eggs so they don’t scramble. Ladle 1 cup of hot broth from the crockpot into the lemon-egg mixture, whisking constantly.
Repeat with a second cup, whisking until smooth and slightly thickened.
Pour the tempered mixture back into the crockpot while stirring. The soup will turn silky and pale. Do not boil after adding the eggs.
Return the shredded chicken to the crockpot.
Stir in olive oil if using. Taste and adjust with more salt, pepper, or lemon juice.
Serve hot topped with chopped dill or parsley. Add extra lemon wedges on the side if you like it brighter.