Add onion, carrot, celery and thyme to a large pot with 10 cups of water. Bring to a boil, then simmer for 30 minutes. Add the chicken and cook for an additional 20 minutes. Remove chicken from the broth and shred. Strain the broth and reserve.
Heat the oil in a large soup pot on medium heat. Add the onion, carrots, and celery. Season with salt and pepper and cook for 5-7 minutes. Stir in the garlic and red pepper flakes and cook for an additional minute. Add the reserved poaching liquid and rice, bring to a boil and then reduce heat to simmer for 10 minutes. Stir in the shredded chicken and cook for an additional 5 minutes.
Whisk the eggs in a separate bowl. Slowly add one ladle full of hot broth to temper the eggs, whisking constantly. When the eggs are tempered, stir them back into the soup mixture. Add the lemon juice, zest and spinach and stir until spinach just wilts. Turn off the heat and stir in the dill. Season with salt and pepper to taste and serve immediately, drizzled with chili oil.
Cook's Tip: The soup thickens as it sits because of the rice. Add a little more broth or water to thin it out.
Heat the oil and red pepper flakes in a saucepan over low heat for 5-7 minutes.