Heat broth, carrots, and celery in a medium pot, covered, over high heat until boiling. Meanwhile, make meatballs.
Combine egg, cumin, 2 tsp. oregano, and the salt and pepper in a bowl. Stir in turkey and rice to blend well. Shape meat into 1 1/2-in. balls and set on a tray.
Lower meatballs into soup with a spoon. Bring to a simmer; reduce heat and cook, covered, until carrots are very tender, 8 to 10 minutes. Stir in remaining oregano.