Heat a large heavy bottomed pot or dutch oven over medium heat. Add the olive oil and heat through then add the carrot, celery and onion and sauté, stirring occasionally, until the vegetables are tender, about 6 minutes. Season the vegetables with salt and pepper then add the garlic and thyme, stirring to combine.
Add the butter and allow to melt then stir in the flour, continuing to stir consistently, until the roux is a deep golden brown, about 2 minutes. Add the chicken stock and bring to a simmer until the broth has slightly thickened, about 30 minutes. Season with salt to taste. Stir in the shredded chicken.
For the dumplings:
In a medium mixing bowl whisk together the flour, baking powder, baking soda and salt. In a separate small bowl whisk together the buttermilk and schmaltz. Stir the buttermilk mixture into the flour with a wooden spoon, until just combined, being careful not to overmix.
Using a spoon, drop generous spoonfuls of the dough into the broth, spacing them out as much as possible (it’s ok if they touch). Cover the pot and reduce heat to low. Cook, undisturbed until the dough puffs up and they are cooked through, about 12 minutes.
Ladle the soup into bowls and top with parsley. Serve hot.