Combine the flour, milk and ¼ teaspoon salt in a food processor. Pulse until the dough forms into a ball.
Roll out the dough on a floured board and cut into 2 ½- by 2 ½-inch squares. Drop about 2 teaspoons of mashed potato in the center of a dough square. Dip a finger in water, run it around the edge of the square, fold the potato inside and pinch the edges to seal. Repeat with the remaining dough and potato filling. Place the finished pierogies onto floured sheet pans.
Bring a large pot of salted water to a rolling boil and drop the pierogies in. When they all float to the top, they are done, 3 to 4 minutes. Strain and place in a bowl.
While the pierogies are cooking, in large skillet over medium heat, melt the butter. Add the onions and saute until medium brown, 6 to 8 minutes. Pour over the pierogies, gently toss and serve.