Make the rolls: Butter a 9-by-13-inch baking dish. Whisk the sugar and cinnamon in a bowl. On a floured surface, roll out the dough into a 10-by-18-inch rectangle. Spread the butter over the dough, leaving a 1-inch border on one of the long sides. Top with the cinnamon sugar. Brush the clean border with water. Tightly roll the dough into an 18-inch log, rolling toward the clean border; pinch the seam to seal.
Slip a long taut piece of thread or unflavored floss under the roll, about 1 1/2 inches from the end. Lift the ends of the thread and cross over the roll, pulling tightly to cut off a piece. Repeat, cutting every 1 1/2 inches, to make 12 rolls. Place the rolls in the prepared baking dish.
Cover the rolls loosely with plastic wrap and let rise in a warm place until doubled in size, about 1 hour, 10 minutes.
Preheat the oven to 350 degrees F. Uncover the rolls and bake until they spring back when pressed, 25 to 30 minutes. Let cool 10 minutes in the pan. (You can freeze the baked rolls for up to 2 weeks. Cool completely before freezing, then thaw, warm up and glaze before serving.)
Make the glaze: Whisk the confectioners' sugar, melted butter, milk, vanilla and salt in a bowl until smooth. Drizzle over the warm rolls.
Warm 1/2 cup water and the milk in a saucepan over low heat until a thermometer registers 100 degrees F to 110 degrees F. Remove from the heat and sprinkle the yeast on top, then sprinkle with a pinch of the sugar; set aside, undisturbed, until foamy, about 5 minutes.
Whisk the melted butter, egg yolk and vanilla into the yeast mixture until combined. In a large bowl, whisk the flour, the remaining sugar, the salt and nutmeg. Make a well in the center, then add the yeast mixture and stir with a wooden spoon to make a thick and slightly sticky dough. Turn out onto a floured surface and knead until soft and elastic, about 6 minutes. Shape into a ball.
Brush a large bowl with butter. Add the dough, turning to coat lightly with the butter. Cover with plastic wrap and let rise at room temperature until the dough is doubled in size, about 1 hour, 15 minutes.
Turn the dough out of the bowl and knead briefly to release excess air; re-form into a ball and return to the bowl. Lightly butter a large piece of plastic wrap and lay it directly on the surface of the dough. Cover the bowl tightly with plastic wrap and refrigerate at least 4 hours or overnight.