To make, heat 2 tablespoons oil in a large Dutch oven over medium-high heat.
Season your chicken with salt and pepper.
When the oil is heated, add some of your chicken to the Dutch oven and cook until browned, about 5 minutes.
Flip the chicken and cook until browned, another 5 minutes. Transfer the chicken to a plate and repeat with your remaining chicken.
While waiting for the chicken to cook, chop your onion, celery, and half of your carrots.
When all of the chicken has been cooked and has been removed from the Dutch oven, add the prepared onion, celery, and carrots to the pot. Season with salt and pepper then lower the heat to medium. Cook, stirring occasionally, for 5 minutes.
Return your chicken to the pot, the add the fresh thyme sprigs. Fill the Dutch oven with 8 cups of water.
Simmer, uncovered for 40 minutes, or until the liquid has reduced by one quarter.
Transfer the chicken to a cutting board to cool.
Take the Dutch oven and pour the liquid through a strainer, reserving the broth. Discard the strained vegetables and wipe out the Dutch oven.
Return the Dutch oven to the stove over medium heat and add in the butter.
When the butter has melted, add in 1/4 cup flour and stir constantly for 2 to 3 minutes, or until it is a pale golden brown.
Slowly whisk in your reserved broth, whisking until no lumps remain. Bring to a boil.
While waiting for the mixture to come to a boil, chop your shallots and remaining carrots.
Add the shallots and carrots to the pot, season with salt and pepper, and lower the heat to a simmer.
Shred the cooled chicken.
Add to the pot and cook, stirring occasionally, for 10 to 12 minutes, or until the liquid has thickened a bit and the carrots are tender.
While the pot cooks, make your dumplings. Add the flour, baking powder, salt and pepper to a medium sized bowl, whisking to combine.
In a small bowl, whisk together the buttermilk and egg.
Add the wet mixture to the dry mixture, along with the melted butter.
Use a wooden spoon to stir the ingredients until just combined.
Using a small spoon, drop generous quarter-size dollops of the dumpling dough into the pot. They will float right on top.
When all of the dough is in the pot, cover, reduce the heat to low, and cook for 20 minutes. After 20 minutes, remove one dumpling and cut it in half to see if it is cooked through. If there are still bits of raw dough, recover the pot and cook for a few minutes more.
To serve, ladle the mixture into bowls and sprinkle with parsley and chives.