Add all of the bread machine ingredients in the order as listed above, into the bread machine.1 1/4 cup whole milk, 1/4 cup butter, 1 large egg, 1 tsp salt, 3 tbsp granulated sugar, 3 cups all purpose flour, 1/2 cup instant mashed potato flakes, 2 1/2 tsp instant yeast
Choose the 'Dough' cycle. Let the bread machine do its thing.
Once the dough cycle has completed, remove the dough and place it on a lightly floured surface, then cover with a clean tea towel. Let rise for 15 more minutes.
After this time divide the dough into 2 equal portions.
Roll each portion into a 12 x 8 inch rectangle.
Brush each rectangle with the melted butter (1.5 tbsp on each rectangle, so 3 tbsp total).
Then cut each of the rectangles in half lengthwise. Now fold each half in 'almost' half lengthwise, leaving about half an inch uncovered. This 'fold' is what give the Parker House Rolls their distinct look. (see image)
Now cut the folded dough into 6 pieces along the long length (see image).
Grease a 9 x 13 baking pan with 3 tbsp of melted butter.
Arrange the cut pieces in the prepared pan with the long sides of the buns running along the long sides of the pan (see image).
Cover the pan with a clean tea towel and set aside in a warm spot to rise for about one hour (I do this part in my oven on the proofing setting).
After this time, preheat your oven to 350 degrees F and bake the buns for 20 minutes.
Once the buns are done, brush with more melted butter and sprinkle with a bit of salt.2 tbsp butter, 1/4 tsp salt
Let the buns cool in the pan for about 10 minutes, then transfer to a wire rack to cool fully.