Melt 1 tablespoon butter in a dutch oven or other heavy bottomed pot with matching lid over medium-high heat. Season the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper, then place in the pot and sear for 3-4 minutes, until browned. Flip the chicken and sear for an additional 3 minutes, or until browned, then remove from the pot.
Melt the remaining 2 tablespoons butter in the pot and add the chopped onion, celery, carrot, and garlic. Saute for 4-5 minutes, until the onion is translucent and the vegetables have started to brown slightly.
Sprinkle the gluten free flour over the vegetables and whisk to combine, then slowly pour in the chicken broth, whisking constantly to get out any lumps.
Stir in the cream, remaining 1/4 teaspoons salt and pepper, sage, thyme, and peas. Add the chicken back to the pot, then cover and reduce heat to medium-low. Let cook for 20 minutes.
While the soup is cooking, make the dumpling batter. In a small bowl, whisk together the flour, salt, baking powder, and baking soda. Then, mix in the milk, butter, and egg until fully combined and set aside.
Remove the chicken from the soup and transfer to a bowl and shred with two forks, then return to the pot.
Drop the dumplings in by the spoonful (about the size of a tablespoon), making sure not to overcrowd the pot as they will double in size. Note that it is very important to make sure the soup is not boiling when you drop in the dumplings or as they cook, or they will fall apart, so if the soup is boiling, reduce the heat to low before proceeding with this step.
Cover the soup and cook for 15-20 minutes, until dumplings are cooked through and have expanded in size. Garnish with fresh parsley and serve!