Place peeled and cubed potato into a saucepan and cover with about 1 inch (2-3 cm) of water. Bring to a boil then turn down and cook covered for about 10 minutes until fork tender.
Scoop out 5 Tbsp (75 ml) of the cooking liquid from the potatoes and drain the rest. Return the potatoes to the pot and cook on medium low to evaporate any excess liquid.
Rice or mash the potato and measure out one cup of packed mash.
To the potato, add the butter and stir to combine and melt the butter. Allow to cool.
To the cooled, reserved cooking liquid add in the sugar and egg and beat to combine until the sugar has dissolved.
In the bowl of your stand mixer, whisk together flour, yeast, and salt.
Add in potato mixture to the flour and combine with hands, breaking up larger clumps until pebble consistency.
With dough hook in place, turn on mixer to low speed and slowly add potato water mixture until there is no dry flour.
Turn up speed on mixer to medium-low and knead for 8 minutes, until the dough clears the sides but the bottom is still sticky.
Tip dough out onto a lightly floured surface and knead by hand about 30 seconds until it is a smooth ball.
Lightly grease a large bowl and place the dough in, turning to coat.
Cover with plastic wrap and let rise 30 - 60 minutes until the dough has doubled in bulk.
Prepare a baking sheet with parchment paper.
Tip out dough on to a clean counter and gently deflate a bit.
Cut into 12 equal weight pieces and place some lightly greased plastic wrap over top.
Working one piece at a time, form into a ball and place seam side down on the parchment-lined baking tray. You should have space for all 12 with a bit of room between.
Once all 12 balls are on the parchment, cover loosely with the lightly greased plastic wrap and allow to proof 30 - 60 minutes until an indent made springs back halfway.
Preheat the oven to 425F / 220C.
Beat the ingredients for the egg wash together in a bowl then brush on to the rolls.
Bake 12 - 14 minutes, rotating the tray once mid-way through the bake. They should be shiny and golden brown when finished.
Remove to a baking rack immediately and cool at least 15 minutes before eating.