Place the potato halves in a 3-quart soup pot and cover with water. Turn heat to high, bring to a boil, and simmer gently for about 20 minutes, or until tender. Drain.
Preheat grill to medium high. Shuck and remove silk from corn. Then dip a paper towel into the oil, rub oil all over the ears of corn, and then place on the grill. Turn every few minutes, until the ears are charred all over, 5 to 7 minutes. Remove from grill and let cool. Cut off the thick end of the cobs so the corn stands upright on your cutting board. Hold the top firmly and slice straight down the sides to remove kernels. Set corn kernels aside.
Melt the butter in a medium skillet over medium heat. Add the onion, celery, and garlic; cook for about 2 minutes or until aromatic. Add the kale, stir, and cook for about 2 minutes, until the kale wilts. Add the potatoes, milk and cream, bring to a boil, and cook for about 5 minutes. Add the oysters with their juices, corn, and stir. Bring just to a boil and the edges of the oysters begin to curl, stirring occasionally. Serve in large hot soup bowls with toasted bread.