Break down chicken and season with salt and pepper. Cook drumsticks and thighs sous vide at 165° for 1 hour or up to 3 hours. Cook breast and wings at 155° for 1 1/2 hours up to 3 hours. Chill in ice water bath for at least 20 minutes. Keep gelatin, if any. Remove skin and discard, remove meat from bones and shred.
Melt butter and olive oil in a pot over medium-high heat add diced onion, carrots, celery, and potatoes. Stir and cook for 3 to 4 minutes over medium-low heat. Stir in ground thyme and turmeric, then pour in chicken broth and apple cider. Stir to combine, bring to simmer and cover pot and simmer for 10 minutes. Add shredded chicken and simmer for another 5 minutes.
While chicken is simmering, make the dough for the dumplings: sift together all dry ingredients, then add half-and-half, stirring gently to combine. Set aside.
Add peas to simmering chicken and vegetables. Pour heavy cream or half and half plus butter into the pot and stir to combine. Check seasonings; add salt and pepper if needed.
Drop tablespoons of dumpling dough into the simmering pot. Add minced parsley. Cover pot most of the way and continue to simmer for 15 minutes. Cook until dumplings are done. Allow to sit for 10 minutes before serving.