Heat oil in heavy large pot over medium heat. Add onions and garlic; saute until translucent. Add turkey wings, chicken broth and oregano. Cover; simmer over medium-low heat for ~1 hour.
Using tongs, transfer wings to work surface; cool. Remove meat from bones; discard skin & bones. Mix meat into soup. Bring to simmer. Add tomatoes with juices, carrots and celery. Cover; simmer until vegetables are crisp-tender for about 6 minutes. (Can be made 1 day ahead. Cool soup slightly. Chill uncovered until cold, then cover. Keep chilled. Bring to simmer before continuing). Add orzo; simmer, covered, until pasta is tender, stirring often (about 10 minutes). Mix in cannellini and basil; cook until heated (about 5 minutes). Season with salt & pepper.