Salt and pepper both sides of chicken thighs. Place chicken thighs into a skillet over low heat. When the fat from the chicken begins to render out cook chicken for 4 to 5 minutes over medium heat. Flip chicken thighs over, and cook for another 4 to 5 minutes. Remove chicken from skillet and set aside.
Add chopped onions, sliced celery, and sliced carrots to the skillet. Sprinkle a small amount of salt over the vegetables. Saute vegetables until the onions are translucent. Add dry Sherry to skillet and scrape the bottom of the pan until the browned bits are no longer sticking to the pan. Place partially cooked chicken thighs, chicken wings into a stock pot. Add vegetables and liquid from the skillet to the stock pot. Pour chicken stock into the stock pot, and cook the soup for about 30 minutes on medium heat.
Remove chicken from stock pot, and allow to cool. Leave the stock pot on the stove over medium heat. While the chicken is cooling, prepare the dumplings. In a medium-sized bowl add flour, sugar, salt, and baking soda. Stir the dry ingredients together. Stir in half and half and melted butter.
Stir the dumpling dough until it forms a ball. Shape dough into golf ball sized pieces and drop them into the stockpot. Debone chicken and place it back into the stockpot. Cook for another 10 to 15 minutes. The dumplings should increase in size, and be fully cooked. If desired sprinkle with fresh parsley before serving.