Heat the oil in a large pot, and sauté the onions until soft but not browned. Add the potatoes, chicken stock and salt. Bring to a boil. Turn the heat down to a simmer and cook until the potatoes are very tender, about 1 hour.
Puree the soup in a food processor, and return to the pot. Bring to a simmer.
Thinly shred the kale (you want it to be very thin). Add to the soup and simmer for 20 minutes.
Just before serving add the oysters to the simmering soup and stir for a few minutes. Season with black pepper.