Place chicken, chopped onion, celery, carrot, bay leaves, 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 bowl to cool, strain all vegetables from simmering broth, discard vegetables, and return broth to the pot. Chop cooked chicken meat.
Combine 2 cups diced onion, olive oil, and a pinch of salt in a large skillet. Cook and stir onion mixture over medium heat until onion is soft, sweet, and golden, 7 to 10 minutes.
Stir onion mixture into chicken 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 eggs, cayenne pepper, and black pepper together in a bowl; whisk in lemon juice. Slowly pour 1 cup broth mixture into egg mixture, whisking constantly. Add 1 more cup of broth to egg mixture and whisk. Pour egg mixture into broth in the pot, add chopped chicken, and cook until heated through, about 5 minutes more.