Place chicken in a pot with root vegetables and bay leaves and season with salt and pepper. Add lemon slices to pot, cover with 1 1/2 to 2 quarts of water and bring to a boil. Reduce heat to medium-low and simmer to poach chicken until cooked through.
Strain the cooking liquid, let chicken cool to handle. Remove skin and chicken bones then pull chicken into small pieces or chop, and reserve.
In a saucepot, combine 1 cup rice with 1 2/3 cups poaching liquid. Bring to a boil, stir, cover and cook to almost tender, 15 minutes. Turn off heat and keep covered.
In a soup pot, heat EVOO over medium-high heat. Add garlic and stir a minute then add spinach and wilt. Add remaining poaching liquid, plus 1 quart stock-in-a-box. Bring to a boil. Whisk eggs with cream or milk, salt, pepper, a few grates of nutmeg and cheese. Drizzle eggs into soup with a fork, the eggs will look like puffy, irregular ribbons running through the soup. Fluff rice with fork, add rice and reserved chicken to soup.