Place water and sugar in a large bowl. You could use the bowl of your stand mixer if you wish.
Add in the yeast and allow the mixture to sit for 5 minutes.
Add in the butter, mashed potatoes, and 2 cups of flour. Mix until completely combined.
If using a stand mixer, switch to your dough hook. Add salt and remaining flour and knead by machine for 4-5 minutes. If kneading by hand, place dough on generously floured surface and knead 7-8 minutes. Dough will be very soft and slightly sticky. It may not come cleanly away from side of the bowl.
Place dough in a greased bowl. Turn to coat the dough and let rise about 45 minutes or until double in bulk.
Place parchment on two baking sheets.
Punch down dough and roll out 1/2 inch thick. Cut with floured shamrock cookie cutter, placing each roll on prepared baking sheet as it is cut.
Cover loosely with a towel and allow rolls to rise for approximately 30 minutes. They won’t be doubled, but should look puffy. If not, give them a few more minutes.
Heat oven to 375 F.
Whisk together the egg and 1 tablespoon water. Brush onto the rolls with a pastry brush and sprinkle with preferred topping – or just leave them plain.
Bake 14-15 minutes, or until rolls begin to turn golden brown.
Remove from the oven and move to a rack to cool completely, or serve warm.
