Stir all the filling ingredients together until well-combined. Set aside.
Stir the dry ingredients (flour, sugar, salt, yeast) together in a large bowl.
Add the butter and milk to a heatproof bowl and microwave until melted but not bubbling. Stir in the water and take a temperature reading with an instant thermometer. You're aiming for a temperature of 110-115º F, which will allow the yeast to bloom. Microwave for a few more seconds or put in the fridge if you need to adjust the temperature.
Pour the liquid ingredients into the dry ingredients, add the egg, and stir with a wooden spoon until thoroughly combined. The dough should be soft and a little sticky.
Lightly flour a work surface and your hands, then turn out the dough. Knead for 3-4 minutes, adding more flour as needed. Don't be afraid to use a lot of flour if you need to. Pat the kneaded dough into a ball.
Coat a bowl with oil or nonstick spray, add the dough, and cover with a clean cloth towel. Let rest for 10 minutes.
Once the dough has rested, roll it out into a 14x8 inch rectangle (dusting the surface and rolling pin with flour) and evenly distribute the filling on top.
With the long side of the dough facing you (think landscape orientation), tightly roll up the dough by folding the far edge over the filling and then rolling it toward you.
Cut the edges off the rolled dough, set them aside, and cut the rest of the log into 12 equal pieces. I did this by cutting it in half, then in quarters, then cutting each quarter in thirds.
Lightly grease a 9" pie pan and arrange the rolls in the pan. Use the edges you cut off to fill in any blank spots. Cover the pan tightly with aluminum foil or a clean shower cap, then set in a warm place to rise for 1.5-2.5 hours, until doubled in size.
Near the end of the rise, preheat the oven to 375º F. Uncover the rolls and bake for 25-30 minutes until they're lightly golden.
While the rolls bake, whisk together the icing ingredients until no lumps of sugar remain. When you remove the rolls from the oven, drizzle the icing on top. The heat allows the icing to permeate the rolls.
Let cool slightly, then serve warm. Store any uneaten rolls at room temperature for up to 2 days.