Heat milk in a microwave-safe bowl until lukewarm, for about 20-30 seconds on high heat. You can also do this in a saucepan on the stove.
In a stand mixer with a dough hook, combine egg, flour, sugar, yeast, salt, butter, and mashed potatoes. Add the warm milk and water mixture and knead for about 5 minutes until the dough is smooth and soft.
Shape the dough into a ball and place it in a large, lightly greased bowl. Cover it with plastic wrap and let it rise in a warm place until doubled in size, about 1 to 1 ½ hours.
After the first rise, divide the dough into 12 equal pieces (about 100g each) and shape each into a smooth ball. Arrange these in a greased baking pan.
Cover the pan and let the rolls rise until nearly doubled in size, about 1 hour.
While the buns are rising, preheat your oven to 350°F (177°C) and get ready to prepare the egg wash.
In a small bowl, crack the egg and lightly beat it with a fork or whisk. Add a tablespoon of milk, a pinch of salt, and a pinch of sugar. Whisk everything together until well combined and slightly frothy.
Use a pastry brush to gently brush the surface of each bun. Be sure to cover the tops and sides for even browning.
Bake the buns in the preheated oven for 25 to 30 minutes, or until they're golden brown on top and sound hollow when tapped on the bottom.
Remove the buns from the oven and let them cool on a wire rack for at least 20 minutes before serving.