Scrub potatoes and place in saucepan, cover with water. I make 3-4 potatoes per person depending on the size. Cover and bring to boil over medium high heat. Remove lid and reduce heat to medium, boil gently until potatoes are pierced easily with a sharp knife. Drain in colander and let cool until they can be easily handled.
Using a heavy bottom measuring cup or other flat surface gently press potatoes to smash, flattening to about 1/2" thickness. The skins will break and the flesh will be exposed, but it should still remain connected. I think this easiest to do on wax paper.
Heat cast iron skillet over medium high heat until very hot. Add olive oil and heat 30 seconds. Add potatoes, fry until golden brown on first side, about 5-7 minutes. Use a combination of tongs and a spatula to gently flip potatoes to second side (this helps keep them from breaking apart). Fry an additional 3-5 minutes until golden on second side.
As the potatoes are finished (which is usually not all at the same time) transfer them to paper towels to drain and sprinkle with finishing salt. Serve immediately.
