Heat olive oil in a large stockpot over medium-high heat. Cook and stir 2 chopped carrots, onion, and celery until onion is golden, 7 to 10 minutes. Pour in wine and bring to a simmer. Add turkey carcass, chicken stock, water, bay leaf, and bouquet garni; bring mixture to a boil.
Reduce heat to a simmer and cook, partially covered, for about 1 hour. Skim any fat that comes to the surface and discard.
Remove turkey carcass and strain soup into a large saucepan over medium heat. Stir in leftover turkey and diced carrot; simmer until warmed through, about 5 minutes. Serve garnished with fresh tarragon.
