Bring carrot juice to a boil in a medium saucepan over high heat. Boil 3 minutes. The carrot solids will "break" away from the clearer liquid during boiling.
Turn off the heat and let stand 5 minutes, allowing the carrot solids to settle to the bottom of the pot. Ladle off the clear liquid. Strain through a double layer of cheesecloth. Discard the solids.
Return the strained carrot juice to the pan. Add the stock, the rice vinegar, and the ginger.
Bring to a simmer over medium-high heat; reduce the heat and simmer for 30 minutes.
Add the chicken and the chile. Reduce the heat to low and steep for 5 minutes more.
Serve, garnished with enoki mushrooms and chives.