In the bowl of a stand mixer fitted with a paddle attachment (please see notes if kneading by hand) mix the potatoes, eggs, butter, milk and water together on medium-low speed until blended.
In a separate medium bowl, whisk the flour, yeast, sugar and salt together, then mix into the potato mixture on medium-low speed until moistened.
Switch to a dough hook and knead the mixture for 10 minutes, until a soft dough has formed. It should be fairly sticky but still clear the sides of the bowl.
Place the dough in a large oiled bowl, turning the dough in the bowl to coat all sides with oil. Cover with cling wrap and let rise in a warm place until the mixture has doubled in size, about 1 ½ hours.
Butter a 9x13 inch (23x33 cm) pan and set aside.
On a lightly floured work surface, divide the dough into 12 equal portions (4 oz/115 g), then roll each piece into a smooth ball.
Arrange 4 rows of 3 balls each seam side down in the prepared baking pan.
Cover again tightly and let rise in a warm place until doubled in size, about 45 - 60 minutes.
Towards the end of the rising time, preheat the oven to 350°F (180°C).
Brush the proofed rolls with the egg wash, then bake for about 30 minutes, until the tops are golden brown.
Let cool for 5 minutes, then serve while still warm or at room temperature. Store leftovers in an airtight container for up to three days. These rolls freeze really well up to 8 weeks.