Preheat the oven to 375 degrees Fahrenheit. Grease a nine-by-13-inch pan with nonstick spray.
PREPARE THE FILLING: In a medium bowl, mix the melted butter with the sugar, flour, cinnamon and vanilla extract to combine. Set aside.
MAKE THE ROLLS: On a lightly floured surface, roll out the dough into a half-inch-thick rectangle. (It should be about 12 inches by 15 inches, but no need to be precise.)
Dollop the cinnamon filling all over the dough, then spread evenly to cover, reaching all the way to the edges.
Starting with the long side of the rectangle that’s closest to you, roll the dough away from you into a tight spiral. When the dough is fully rolled, use a serrated knife to slice the dough into nine even pieces (each about one-and-a-half inches thick).
Transfer the cinnamon rolls to the prepared pan, leaving space between them for the dough to expand.
BAKE THE ROLLS: In a small bowl, whisk the egg and water to combine. Brush the egg wash onto the rolls, then transfer the baking dish to the oven. Bake until the rolls are golden and the filling is bubbly, 30 to 40 minutes. Let cool for 15 minutes.
MAKE THE ICING: In a medium bowl, whisk the confectioners’ sugar with the milk and vanilla extract to combine. Drizzle the frosting over the slightly cooled rolls. Serve warm, immediately. Cooled rolls can be stored for up to two days and reheated at a low temperature in the oven (or in the microwave).