Heat the olive oil and butter in a 5 1/2-quart Dutch oven over medium-high heat. Season the chicken with salt and pepper on both sides and place skin-side down in the pot. Brown for 4 to 6 minutes on the first side, then flip and brown for 4 to 6 minutes on the other side. Remove to a plate and set aside.
Add the mushrooms, garlic, bell peppers, onion, thyme and turmeric to the pot. Cook, stirring occasionally, until the vegetables are softened and the edges are starting to brown, about 2 minutes. Add the tomato paste and cook for an additional minute. Pour in the white wine, bring to a simmer and simmer for 3 minutes. Use your spoon to scrape the sides and bottom of the pot to loosen any browned bits that might be stuck there. Add the chicken stock, diced tomatoes and chicken thighs. Season the soup with a healthy pinch of salt and pepper. Bring to a boil, reduce to a simmer and cook, uncovered, until the chicken is cooked through and the vegetables are tender, about 20 minutes.
Fill a 4-quart pot with water and bring to a boil. Cook the egg noodles until al dente according to the package directions. Drain the noodles, put back into the pot they were cooked in and toss with a scant tablespoon of olive oil; set aside.
When the chicken is ready, transfer it to a cutting board. Using 2 forks, remove the chicken from the bones and shred. Some skin will end up in the mix, which is totally fine. Add the shredded meat back to the soup. Discard the skin and bones.
Add the noodles to the soup, as well as the heavy cream and 2 tablespoons of the parsley. Simmer for a few minutes to heat everything through. Taste the soup and adjust the seasoning as needed. Serve hot in bowls and garnish with the remaining chopped parsley.
If you are making this soup ahead of time, undercook the noodles by 2 minutes as directed on the package, so they don't overcook when the soup is reheated. Alternatively, cook the noodles, drain, toss in olive oil and store separately, adding them to the soup once it has been reheated. Serve as directed above.