Pat chicken dry with paper towels and season with salt and pepper. Heat oil in dutch oven over medium heat until just smoking. Cook chicken, skin side down, until well browned, 6-8 minutes; transfer chicken to plate. Remove and discard skin.
Pour off all but 1 tsp fat & return pot to medium heat. Add onion & cook until just beginning to brown, 3-5 minutes. Add broth, chicken, and 1-1/4 tsp salt; bring to boil. Reduce heat to medium-low, cover, & simmer until chicken is tender, about 30 minutes.
Meanwhile, whisk flour, baking powder, 1 tsp salt, & 1/2 tsp pepper together in a large bowl. Whisk eggs, 1/2 cup half-and-half, and water into flour mixture until thick batter forms. Transfer batter to gallon-size zipper-lock bag and refrigerate until ready to use.
Remove chicken from pot and set aside. Add potatoes to broth and simmer 10 minutes. Cut 1/4-inch off corner of bag of batter. Pipe batter into simmering brother, snipping dumplings with moistened kitchen shears every 1/2 to 3/4 inch as extruded. Simmer until dumpling floor to surface, 10-15 minutes, stirring occasionally.
Shred chicken into bite size pieces, discarding bones and add to soup. off heat, stir in remaining 1/2 cup half-and-half. Season with salt and pepper to taste and serve.