Make the preferment: Stir the yeast into the water in a glass measure and let it stand for 5 - 10 minutes. Add ⅓ cup of this yeasted water (discard the rest) to the flour and beat this very sticky starter until it is well combined. Cover with plastic wrap and let it ferment until it is full of huge bubbles and sharp tasting, about 12 hours. If your kitchen is very warm and the pre-ferment is fermenting very quickly, place it in the refrigerator after 3 hours of fermenting. In the morning, remove it and allow it to come to room temperature 30 minutes to an hour before beginning the final dough.
Prepare the potato: You can prepare the potato the night before or just before mixing the dough. Quarter it, then boil it in water to cover until it can be easily pierced with a knife tip, about 20 minutes. Drain; if desired, reserve the water for the dough. Peel the skin off and press the potato through a ricer or sieve, or mash it thoroughly with a fork. Store it in a covered container in the refrigerator. You will need only ¼ cup of potato puree.
Mix the dough: In the bowl of the stand mixer, stir together the flour and water into a rough, very wet dough. Cover the bowl and let it rest (autolyse) for 10 to 20 minutes.
Attach the dough hook. Add the pre-ferment, honey, potato and salt and the mix the dough on low speed (2 on the KitchenAid) for 15 - 20 minutes, or until very silky and wraps around the hook and cleans the bowl before splaterring back around the bowl. This dough is almost pourably wet.
Rising: Shape the dough into a ball and roll it in flour. Place it in a container at least 3 times its size and cover tightly with a lid or plastic wrap. Let it ferment until doubled in bulk and filled with large air bubbles, about 4 hours. Using plenty of dusting flour, turn the dough 4 times in 20 minute intervals, that is, after 20, 40, 60, and 80 minutes of fermenting, then leave the dough undisturbed for the remaining time. Do not allow this dough to over ferment or ferment to the point of collapse, because the flavor and structure of your bread will suffer.
Shape and Proof the Dough: Turn the fermented dough out onto a well floured work surface, shape it into a round and let it rest for 20 minutes. Sprinkle a couche or tea towel or wooden board generously with flour. Slip a baking sheet under the couche/towel if you are using one for support. Sprinkle a couple tablespoons of flour over the center of the ball. Push your fingers into the center to make a hole, the rotate your hand around the hole to widen it, making a large 4 inch opening. I started using both hands to shape the hole just right. The bread should have about 12 inch diameter. Place the dough smooth side down on the floured couche or board and dust the surface with more flour. Drape it with plastic wrap and let it proof until it is light and slowly springs back when lightly pressed, about 1-½ hours.
Preheat the Oven: Immediately after shaping the bread, arrange a rack on the oven’s second to top shelf and place a baking stone on it. (The dusted flour burned, so I will not move the rack up next time.) Clear away all the racks above the one being used. Preheat the oven to 450° F (230° C).
Bake: Unwrap the bread and flip it onto a floured peel or a sheet of parchment paper. Do not worry about damaging the bread as you handle it; it will recover in the oven as long as it is not over-proofed. Slash it with 4 radial cuts in the shape of a cross. Slide the loaf onto the hot baking stone and bake until it is very dark brown, 40 -50 minutes, rotating it halfway into the bake. Let the bread cool on a rack.