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. 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. Mash potatoes.
Place 1/2 cup potato cooking water in a large bowl or the bowl of a stand mixer fitted with the paddle attachment. Add 1/2 cup mashed potatoes. 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 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 the flour gradually, until you have a moist dough ball, not sticky to touch, but almost. It should clean the bowl a bit and wrap the dough hook.
Remove dough to a floured work surface and knead briefly, adding a bit of additional flour only if the dough is sticking to your hands or the work surface. Form dough into a ball and place into a greased bowl, cover with plastic wrap and allow to rise until doubled, about 45-60 minutes.
Preheat oven to 350F (regular bake/not fan assisted). *Reduce oven temperature to 325 F if using a glass baking pan. 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. Form each piece into a ball 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. Cool rolls in the pan for 5 minutes, then remove to a cooling rack to cool to warm or to cool completely. If not enjoying immediately, store in an airtight container or freeze.
