In a large bowl, combine the cooled mashed potatoes, flour, baking powder, and salt. Mix until a dough forms.
On a floured surface, turn out the dough and knead gently for about 1 minute until smooth.
Divide the dough into four equal pieces. Roll each piece into a circle about ½ inch thick. Cut each circle into quarters to form wedges.
Heat a skillet over medium heat and add a little butter. Fry the potato farls for about 4-5 minutes on each side, or until golden brown and crispy.
Remove from the skillet and keep warm while you cook the remaining farls.
Serve warm with butter, or enjoy with your favorite toppings.