To prepare the meatballs, preheat the oven to 400°F. Line a baking sheet with parchment paper.
Combine the chicken, onion, garlic, Parmigiano-Reggiano, breadcrumbs, mayonnaise, egg yolk, Italian seasoning, parsley, salt, and pepper in a medium-size bowl. Using clean hands, mix thoroughly until well combined.
Form the chicken mixture into small balls, no more than 1'' round. They should be small. Place the meatballs on the prepared baking sheet. Bake until cooked through, 5-8 minutes. Let the meatballs cool on the baking sheet.
To prepare the soup, heat the olive oil in a large stockpot or Dutch oven. Add the onion, carrots, and celery, and sauté just until soft, 6-8 minutes. Add the chicken broth, thyme, and bay leaves and bring to a simmer. Simmer for 15-20 minutes. Add the orzo and simmer until tender but still firm to the bite, 6-8 minutes. Add the meatballs and spinach to the soup and heat through, 5-8 minutes. Remove and discard the bay leaves. Ladle the soup into warm bowls. Garnish with fresh Parmigiano-Reggiano, if desired, before serving.
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