Heat the olive oil in a large pot over medium heat. Add the chicken pieces and cook until lightly browned on all sides. Remove the chicken and set it aside.
Add the butter to the same pot. Stir in the onion, carrots, and celery. Cook for 5 minutes until the vegetables begin to soften. Add the garlic and cook for 1 minute more.
Sprinkle 1/4 cup of the flour over the vegetables and stir for 2 minutes to form a light roux. Slowly pour in the chicken broth while stirring to avoid lumps. Return the chicken to the pot and simmer for 10 minutes.
Stir in the heavy cream and season with salt and black pepper. Reduce the heat to low and let the mixture gently simmer.
In a mixing bowl, combine the remaining 1 1/4 cups flour, baking powder, and 1/2 tsp salt. Pour in the milk and stir until a soft dough forms. Drop spoonfuls of the dough directly into the simmering liquid.
Cover the pot and cook for 15 minutes without lifting the lid so the dumplings steam properly. Stir in the fresh parsley and thyme just before serving.