Place potatoes in a large saucepan and cover with cold water. Bring water to a boil, then reduce heat to medium and cook until potatoes are tender, about 20-25 minutes. Pour 1 1/2 cups of the potato water in to a measuring cup and set aside. Drain /discard remaining potato cooking liquid. While potatoes are cooking, bring milk to a simmer in a small pan, then remove from heat.
Mash or rice the hot potatoes and place in a large bowl or the bowl of a stand mixer fitted with a paddle attachment. (You should have about 3 cups of mashed potatoes). Add the sugar, salt, and butter and blend well. Add the reserved potato water and the milk and beat well. Allow mixture to stand until it cools to lukewarm.
Add the yeast to 1/4 cup warm water, stir, and set aside for five minutes until bubbly. Gradually add four cups of flour to the potato mixture and mix well. If using a stand mixer, switch to the kneading hook. Add the dissolved yeast and blend well. Add the remaining flour, about 1/2 cup at a time, mixing well after each addition, until the dough is stiff enough to knead. You may not need to use all of the flour.
Turn dough out onto a floured board and knead until smooth and elastic (about 5 minutes), adding additional flour if necessary. Brush a large bowl with a thin layer of oil and place the dough in the bowl. Turn dough to coat with oil, cover with plastic wrap, and let rise for 60 - 90, or until doubled in size.
Preheat the oven to 400° F. Lightly grease two 9x13-inch rimmed baking sheets and lightly dust with a bit of flour. *If you don't have two baking sheets that will sit side-by-side on one oven rack, position oven racks in top and bottom third of oven and bake on two racks, switching racks half way through baking time. You can also use two 9x13 glass baking dishes, if that's all you have.
Turn the dough out onto a floured board, gently punch it down. Divide dough in to two equal pieces. Take one of the dough pieces and cut in half again, to make 2 pieces. Cut each of those 2 pieces in half again, to make 4 pieces. Then cut each of those pieces in to 3 or 4 irregular pieces, to make 12-16 pieces. Arrange those pieces in a cobblestone pattern, leaving a bit of space between each and filling the entire pan as much as possible. Repeat with the other dough half and fill the second baking sheet in the same way. Lightly dust the top of the rolls with flour, cover with a clean tea towel and then let rise until doubled in size, approximately 30 minutes.
Bake with the baking sheets side-by-side on one (centrrack in the preheated oven for about 20 minutes, until golden brown on top. *At the 10 minute mark of baking, you may want to switch the pans to the other side and front to back, to ensure even browning. Serve warm or at room temperature. These rolls freeze well. Thaw in wrapping at room temperature.