Melt buttter in a saucepan over medium heat. Add onion and cook, stirring often, until onion is translucent. Add chicken in one layer; sprinkle with salt and pepper. Add chicken broth, adding water if necessary to completly submerge the chicken. Bring to a boil and cook for five minutes; cover, remove from heat and allow to sit undisturbed for 1 hour. Remove chicken, coarsely chop, and refrigerate for at least 30 minutes to chill. (Save the leftover, delicious, rich chicken broth for another use; we like to make chicken noodle soup with it.)

Slice hard-cooked eggs in half lengthwise; carefully remove yolks, keeping egg white halves intact.

Mash or finely crumble egg yolks. Stir together yolks, mayonnaise, green onions, parsley, dill, and lemon juice; set aside.

Pulse cooled chicken in a food processor 3 to 4 times or until shredded; stir into mayonnaise mixture until blended. Spoon chicken mixture evenly into egg white halves. Cover and chill at least 1 hour.

To serve, sprinkle lightly with smoked paprika.