Make the ricotta dumplings: In a medium bowl, combine all the ingredients, except the flour, until smooth, then transfer to a piping bag. Spread 1½ cups of the flour evenly onto a parchment-lined sheet pan. Pipe the cheese mixture into 12 golf ball-sized dumplings, 1½ inches in diameter, over the flour. Roll the dumplings in the flour until fully coated and cover with the remaining 1½ cups of flour. Wrap the sheet pan in plastic wrap and refrigerate overnight. The next day, shake the ricotta dumplings gently, one by one, in a fine-mesh sieve to remove any excess flour. Reserve for later.
Prepare the broth: Preheat the oven to 400º. Break down the chicken: Remove the breasts, dice and reserve for later, then separate the drumsticks and thighs. Roast the bones, drumsticks and thighs until golden brown, 45 minutes. In a heavy-bottomed stockpot, add the roast chicken and the remaining broth ingredients, and cover completely with water. Bring the stock up to a boil, reduce to a simmer and cook for 1½ hours. Strain through a fine-mesh strainer and reserve the liquid.
Prepare the chicken and dumpling soup: Using the same pot, heat the olive oil and sweat the leeks until translucent, 6 to 7 minutes. Add the chicken broth and bring to a boil; reduce the heat to a simmer, add the diced chicken breast and cook until it reaches an internal temperature of 165º, 10 to 12 minutes. Remove the chicken from the pot and set aside. Add the ricotta dumplings to the broth and cook until tender, 10 minutes. Add the diced chicken back to the pot and stir. Ladle the soup into a bowl, garnish with a sprinkle of ginger and a dill sprig, then serve.