To assemble the dough -- In a large mixing bowl, mix the 3 Tbs. flour with the yeast and then whisk in the water. Let the mixture sit covered until it has begun to puff up, 10 to 15 minutes.
Fit a large-capacity food processor with the metal blade. Put the remaining flour in the workbowl and then add the yeast mixture, potato, egg yolks, vanilla, and sour cream. Process the dough for about 1 minute.
Remove it from the machine and knead it by hand on an unfloured countertop for 1 minute to redistribute the heat. The dough will be very stiff at this point. Continue this alternating kneading: process for 30 seconds and then knead on the counter for about 30 seconds, until the dough is very smooth (this should take 2 to 3 processing rounds).
Put the dough back in the food processor and add the sugar and salt, kneading again in the processor and then on the counter until the sugar has dissolved (the dough will soften considerably and become very sticky; this is okay).
Finally, return the dough to the processor, add the butter, and do another alternating kneading round until the butter is well incorporated and the dough is very soft and smooth, about 1 minute.
The dough won't clean the bowl at this point. It's okay if it feels quite soft and warm after processing: kneading the dough on the counter will help it cool down and firm up.
Transfer the dough to a container at least four times its volume (no need to grease the container); seal well. (At this point, the dough can instead be rolled in flour and then sealed in a plastic bag and refrigerated for up to 4 days. If you do mix ahead and chill the dough, pull it out of the fridge 3 to 4 hours before baking.)
Let the dough ferment at room temperature for about 3 hours or until it's expanded to 3 times its volume and an indent remains when you press it with a floured finger.
For the Chrysanthemums: Put the cinnamon sugar on a large plate and line a heavy baking sheet (or an insulated sheet or two sheets sandwiched together) with parchment or butter it.
Roll the dough into an 8x20-inch rectangle. Cut it into 2x8-inch strips. Make sure the melted butter is quite warm. Dip each strip of dough into the butter, covering both sides, and then dredge it in the cinnamon-sugar on both sides, coating the strips evenly. Using a bencher or a knife, cut a deep fringe the entire length of each strip, spacing the cuts about 1/4 inch apart.
Roll the strips up to form the chrysanthemums, pinching the bottoms to seal them. Position them upright on the baking sheet, spreading and arranging the petals attractively.
Position an oven rack on the top spot and heat the oven to 400°F. Cover and proof the shaped dough until it's large, puffy, and remains indented when lightly pressed with your fingertip, about 1/2 hour.
Bake until brown, about 15 minutes, rotating the pan halfway through baking. Let cool on a rack.