Peel and cut the potatoes into chunks, then add them to a medium saucepan. Cover them with water by at least an inch.
Cook until tender, about 10 minutes. Drain potatoes and reserve about ⅔ cup of the water. Mash them and set them aside to cool.
Meanwhile, in a microwave-safe medium bowl, combine the milk, butter, sugar, and salt and microwave for about a minute. Stir until everything melts.
Add the reserved potato water. If the mixture is too hot, let it cool enough not to kill the yeast and curdle the egg.
Transfer the mixture to a large bowl or stand mixer. Add the instant yeast and egg and whisk until thoroughly combined.
Add the mashed potatoes and continue mixing. Next, add the all-purpose flour a cup at a time. Add just enough flour to make a soft dough.
Transfer the dough to a lightly floured surface and knead for 6-8 minutes.
Place the dough in a greased bowl, turning once to coat the dough. Cover loosely with a clean cloth and let it rise in a warm place for 1½ to 2 hours or until doubled in size.
Punch the dough down and divide it in half. Flatten one half into a rectangle, then fold in thirds on top of itself. Fold the bottom third down so that it overlaps the other layers.
Pinch the dough to seal where the layers meet to close it up. Repeat with the other half.
Lift both loaves and place them into two greased 9"×5" loaf pans. Allow them to rise for about 30 minutes or until the dough has risen an inch above the pans.
Bake at 350℉ (175℃) for 30-40 minutes. If the bread is browning too fast, tent with foil paper. You can easily check for doneness by using an instant-read thermometer. It should read 195-200°F (90-95℃) when inserted into the center of the bread.
Take the loaves out of the oven and transfer them to a rack. Let them cool before slicing.