In a pot, cover the 4 medium peeled and halved potatoes with water and bring to a boil over high heat.
Simmer on medium-high heat until the center of the potatoes are tender when pricked with a fork, about 20 minutes. Turn off heat.
Drain the potatoes, return them to the pot, and allow them to completely dry out over the remaining heat. Mash with a potato masher until smooth.
Place the warm mashed potatoes in a medium bowl. Stir in 1/4 cup of all-purpose flour, 1 pinch of salt, and 1 tablespoon of melted butter. Mix lightly until dough forms.
On a well-floured surface, knead the dough lightly. The dough will be sticky.
Use a floured rolling pin to flatten the dough into a 9-inch circle, about 1/4 inch thick. Cut into quarters using a floured knife.
Sprinkle a little flour into the base of a skillet and cook the farls for 3 minutes on each side, or until evenly browned.
Season with a little salt and serve the Irish Potato Farls straight away.