Boil the Potatoes: Place the 500 g peeled, diced potatoes in a medium saucepan and cover with cold water. Bring the water to a boil, then continue cooking until the potatoes are soft, about 15 minutes.
Mash the Potatoes: Drain the cooked potatoes thoroughly, return them to the saucepan, and add 30 g salted butter. Mash until completely smooth, then whip the mash briefly with a fork to give it an especially light, fluffy texture.
Form the Dough: Add 1 cup self-raising flour to the mashed potatoes. Gently bring the mixture together using a wooden spoon, or use a mixer on low to medium speed. Work just until a light, pliable dough forms that does not stick to your hands. The amount of flour may need slight adjustment depending on the moisture in the potatoes. Roll the Dough: Transfer the dough to a lightly floured work surface. Lightly dust a rolling pin with flour, then gently roll the dough to approximately 1 cm (½ inch) thick.
Cut the Scones: Divide the rolled dough into four quarters, creating four tattie scones.
Heat the Pan: Place a large, heavy frying pan, such as a cast-iron or wrought-iron pan, over medium heat. Add the 1 tbsp butter and allow it to melt for cooking the scones.
Cook the Scones: Place as many scones in the pan as will fit comfortably while still leaving enough room to turn them. Cook for 3–5 minutes per side, turning when needed, until both sides are crispy and golden brown.
Serve Hot: Serve the tattie scones immediately while hot. Enjoy them simply with butter, with jam, or as part of a full Scots breakfast.