Place the chicken in a medium pot, and add enough cold water to just cover the chicken. Bring to a boil over medium-high heat, uncovered.
Once boiling, add the onion and ginger to the pot. Reduce the heat to medium, cover, and cook for 40 minutes. Remove the lid occasionally to skim any foam off the top and discard.
Remove from the heat. Move the chicken to a plate to cool. When cool enough to handle, shred the chicken. Return the chicken carcass and 1-2 cups of the shredded chicken back to the pot. (Save the remaining shredded chicken for another use.)
Add the carrots and dates to the pot, and simmer over low heat, uncovered, for 45 minutes. Add the gogi berries and cook an additional 15 minutes. Season with salt to taste.