Cut the potatoes in half or quarters and place in a medium-large saucepan. Cover potatoes with at least an inch of cold water. Bring to a boil over high heat, then lower heat slightly and continue cooking until potatoes are tender. DO NOT DRAIN YET! Place a strainer over a medium bowl and pour potatoes and water over, reserving the potato water in the bowl and the potatoes in the strainer.
Prepare the bread:
Place 1/2 cup potato cooking water in a large bowl or the bowl of a stand mixer fitted with the paddle attachment. Scoop out the potato flesh, discarding the skin. Place in a bowl and mash with a fork. Measure out 1/2 cup of mashed potatoes and add to the bowl with the potato cooking water. Add the butter (or oil) and honey (or sugar).
Using the paddle attachment, mix on low speed until mixture has cooled to lukewarm (about 120F for instant yeast or 110F for dry active yeast). Sprinkle in yeast and mix to combine. If using instant yeast, proceed to next step. For active dry yeast, let stand 5 minutes before proceeding.
Add salt and 1 cup of the flour to the mixture and still using the paddle attachment, mix until combined. Remove the paddle attachment and replace with the kneading hook. Begin adding the more of flour gradually, until you have a moist dough ball, not sticky to touch, but almost. You may need to add a bit more flour than specified, but be careful you don't add too much. You want to stop when the dough is just not sticky, but is still very moist.
Remove dough to a greased bowl, cover with plastic wrap and allow to rise until doubled, about 45-60 minutes.
Preheat oven to 350F and lightly grease an 8-inch square baking pan. Set aside.
Remove dough to a floured work surface and gently deflate. Divide dough into 9 equal pieces (*see Note 1). Form each piece into a ball (*see Note 3) and place into your prepared pan in 3 rows of 3 rolls, leaving a little space between rolls. Cover with a clean tea towel and set aside to rise until doubled and puffy. Using a fine mesh strainer, dust the top of the rolls with a dusting of flour.
Bake rolls for 20-23 minutes, or until just lightly golden on top and not yet browning on the sides. If you have a thermometer, remove when the centre roll is about 172-175F. Cool rolls in the pan for 5 minutes, then remove to a cooling rack to cool to warm (to enjoy right away) or to cool completely. If not enjoying immediately, store in an airtight container or freeze.
1. I like to weigh the entire dough ball, then divide that weight by 9, to figure out the weight for each roll. That way all the rolls will be the same size and cook at the same rate, as well. 2. If you'd like these to be vegan, opt for the oil instead of butter and the sugar instead of honey. 3. To form the dough into balls, take the piece of the dough and using both hands, stretch dough out from the 3 and 9 o'clock position and tuck underneath, pinching lightly. Repeat from the 12 and 6 o'clock position. Flip over and pinch well where the dough meets underneath. This method creates a tight ball, with surface tension on top, which makes them rise nicely! Be sure to read the "Cook's Notes" in the original post, for more tips, options, substitutions and variations for this recipe!