In a large bowl, combine ground beef, ground chicken, finely chopped onion, minced garlic, parsley, breadcrumbs, egg, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Mix well until all ingredients are evenly incorporated.
Form the meat mixture into small dumplings, about 1 inch in diameter, and set aside on a plate.
In a large pot or Dutch oven, heat olive oil over medium heat. Add diced onion, sliced carrots, and celery. Sauté for 5 minutes until vegetables begin to soften.
Add minced garlic, dried thyme, and dried oregano to the pot. Cook for an additional 1 minute until fragrant.
Pour in the chicken broth and bring the mixture to a gentle boil.
Carefully drop the prepared beef and chicken dumplings into the boiling broth. Reduce heat to a simmer and cook uncovered for 20 minutes, or until dumplings are cooked through and vegetables are tender.
Add the chopped spinach leaves to the pot and stir. Cook for 2-3 minutes until spinach is wilted.
Taste the soup and season with additional salt and black pepper as needed.
Ladle the soup into bowls and serve hot.