Cut the chicken into 8 pieces and season all over with salt.
Heat a large Dutch oven or pot over medium heat and melt the butter. When the butter has melted, and working in batches as necessary, add the chicken pieces skin side down and cook until the skin is golden brown, 7 to 8 minutes. Flip and continue cooking until a beautiful sunshine color develops on the second side, 3 to 4 minutes. With tongs, transfer the chicken to a rimmed plate.
Add the onions, carrots and celery to the pot, season with a pinch of salt and cook until softened and translucent, about 6 to 8 minutes. Stir in the garlic and cook for 45 seconds, until fragrant.
Add the stock, water, chicken, 1 tbsp of the summer savory, and a pinch of salt, and bring to a boil. Reduce the heat to medium, and cook until the chicken is tender, about 30 to 35 minutes. Add the potatoes, and continue cooking for about 12 to 15 minutes, or until the potatoes are just fork-tender. Remove from the heat.
Using a slotted spoon, remove the chicken and transfer to a cutting board. When cool enough to handle, shred the meat with 2 forks, discarding the skin and bones and then return the meat to the broth. Skim and discard the fat on top of the broth. Taste and adjust the seasoning as necessary, adding more salt or pepper.
In a medium bowl, combine the flour, baking powder, the remaining ½ tablespoon of the summer savory and lemon zest. Season with salt and mix well. While constantly stirring, gradually add about ½ to ¾ cup of water to the flour mixture until the dough comes together.
Return the chicken stew to the stove and bring to a simmer over medium heat. When simmering, drop tablespoon-sized portions of the dough into the fricot, cover, and cook for 7 minutes, or until the little dough balls are cooked through.
Divide your fricot into individual bowls, garnish with parsley, and enter a wonderful world of Acadian comfort.