To make the dough: Whisk together the egg and warm water. Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Add the remaining dough ingredients, stirring to combine, then knead until the dough is smooth and elastic, 5 to 10 minutes., Place the dough in a lightly greased bowl, cover with plastic wrap, and let it rise until doubled in bulk, about 2 hours., Turn the dough out onto a lightly greased work surface or silicone rolling mat. Pat, then roll it into an 18" x 10" rectangle. Spread with the pesto and sprinkle on the Parmesan, leaving 1" free of filling along the long edge farthest from you., Starting with the long edge closest to you, roll the dough into a log, pinching the seam closed. Cut the log into 12 slices., Place the rolls on a piece of parchment (if you're using a pizza stone), or on a parchment-lined or lightly greased baking sheet., Cover and let rise for about 1 hour, or until puffy. Sprinkle with additional cheese, if desired. Towards the end of the rising time, set a baking stone in the middle of the oven and preheat the oven to 350°F., Transfer the rolls on the parchment to the baking stone (or the pan to the oven), and bake for 22 to 26 minutes, or until golden., Remove the rolls from the oven, and serve immediately. Or place them on a rack to cool, then wrap airtight and store at room temperature for several days; freeze for longer storage.