Heat 2 tablespoons olive oil in a large pot. Sprinkle chicken pieces with salt and pepper and brown in batches: half of the chicken at a time, browning both sides well. Add additional 2 tablespoons olive oil in the pot to brown the second batch of chicken.
Remove chicken and sauté onion and carrots until the onions start to become tender, 3 minutes or so. Add chicken back into the pot and cover with chicken broth. Add in thyme, poultry seasoning, tarragon, garlic powder, bay leaves, and salt and pepper to taste.
Cover and bring to boil. Reduce heat to simmer and cook for 40 minutes or until chicken is completely cooked through.
Remove chicken pieces and shred, discarding extra fat and bones. Add chicken pieces back into the pot.
To make the dumplings, whisk flour, salt, baking powder, and baking soda together until well mixed. Pour in melted butter and buttermilk and stir until just combined. Drop by the large tablespoon into the simmering pot. Dumplings will be done when they are puffed and floating, after about 3 minutes.
Taste broth and adjust seasonings as needed. Serve hot.