Gather all ingredients.

Place chicken, 1 chopped onion, celery, carrot, bay leaves, salt, and oregano in a large pot. Pour water over the top; bring to a boil, reduce heat to medium-low, and simmer, skimming foam as necessary, until chicken is falling off the bone, 60 to 90 minutes. An instant-read thermometer inserted into the thickest part of the thigh, near the bone, should read 165 degrees F (74 degrees C).

Transfer chicken to a large bowl to cool. Strain all vegetables from simmering broth, discard vegetables, and return broth to the pot.

Chop chicken meat into bite-sized pieces and set aside.

Combine 2 cups diced onion, olive oil, and a pinch of salt in a large skillet over medium heat. Cook and stir onion mixture until onion is soft, sweet, and golden, 7 to 10 minutes.

Stir onion mixture into broth in the pot and set over medium-low heat; add Arborio rice and a pinch of salt. Bring to a simmer; cook, stirring occasionally, until rice is very tender and broth flavors combine, about 45 minutes.

Whisk together eggs, black pepper, and cayenne pepper in a medium bowl; whisk in lemon juice until combined. Slowly pour 1 cup broth mixture into egg mixture, whisking constantly.

Whisk another 1 cup broth mixture into egg mixture, then pour egg mixture into the pot. Stir well.

Add chopped chicken and cook until heated through, about 5 minutes more.

Serve hot and enjoy!