Chop the carrots.
Chop the onions.
Chop the celery.
Put the olive oil and butter in a deep skillet and heat up to medium high heat.
Put the chicken pieces into the skillet and fry the chicken until it is brown on all sides, about 15 minutes. Remove and set aside.
Add the carrots, onions and celery and cook for 3 minutes, stirring to combine the flavors.
Add the tomatoes.
Add the fresh tarragon.
Add the turmeric and salt, and cook for an additional 3 minutes stirring to make sure the mixture does not burn.
Add the chicken stock to the mixture and bring to a boil.
Turn down the heat to a simmer and add the chicken back to the pot for 20 minutes.
Chop the parsley.
Remove the chicken and let cool.
In a separate bowl add the 1/2 cup of half and half and two tablespoons of flour stirring to get rid of any lumps, and add to the pot.
Sift the flour.
Sift the cornmeal.
Sift the salt.
Add the baking powder.
Add 1 1/2 cups of milk to the flour mixture and stir to make the dumplings.
The dumplings should be sticky and if you need to add more flour, do so one tablespoon at a time.
Once cooled shred the chicken using 2 forks, removing most of the skin and all the bones and return to the pot.
Using two spoons, scoop a tablespoon of batter into the soup.
Cover and cook on medium high for 15 minutes.
Remove the pot from the stove and let sit for 10 minutes and serve in soup bowls.