Halve the potatoes and place cut side down, halve each half again but keep these halves together.
Choose a cast iron skillet large enough to accomodate all the potato halves. I wound up making it in two batches. Add enough olive oil to coat the bottom of the cast iron pan- to no more than 1/8 inch deep.Heat the oil until it begins to shimmer. Sprinkle the kosher salt over the olive oil as evenly as possible. Place the potato halves on top of the salt. Keep cooking at medium high heat for 12 to 15 minutes. Check the bottom of a potato to see if they are browning to your liking.
When the poitatoes are nicely browned, turn the heat as low as possibloe and cover the pan. You will hear splattering noises as the potatoes start to steam. Cook for 20 to 25 minutes or until potatoes pierce easily. Serve hot, keep covered. Kept covered, they will stay hot for 30 minutes. If you are letting them stand, drain any oil left in the bottom of the pan.