Make the dough: In a small saucepan, warm the milk over low heat until lukewarm (110-115°F). Pour the warm milk into the bowl of a stand electric mixer fitted with the dough hook attachment. With a spoon, manually stir in the sugar and yeast. Cover with a towel and let sit until the yeast is foamy, about 5-10 minutes. On low speed, beat in the softened butter until it is slightly broken up. Next add the eggs, one at a time, and then the salt. The butter won't really be mixing into the mixture, so don't be alarmed if it stays in pieces. On low speed, gradually add the flour. Once it is all added, beat on medium speed until a soft dough forms. Increase speed to medium-high and beat until the dough is soft and supple, about 6 minutes longer.
Transfer the dough to a lightly floured surface and knead it with your hands for 1 minute. Form the dough into a ball and transfer it to a lightly greased bowl. Cover the dough loosely with plastic wrap, a paper towel, or aluminum foil and let sit in a warm place until doubled in size, about 1 to 2 hours.
Butter/grease/spray with nonstick spray the bottom of a 9x13-inch baking dish, then line with parchment paper. Turn the dough out onto a lightly floured work surface and, using a rolling pin, roll into a 13x18-inch rectangle. You may use a ruler for accuracy. Make sure the dough is smooth and evenly thick, even at the corners.
Make the filling: Using a handheld or stand mixer fitted with a paddle or whisk attachment, beat the butter and maple syrup together on medium-high speed until combined and creamy. Add the sugar and cinnamon, beating until combined. Spread the maple mixture all over the dough. Tightly roll up the dough to form an 18-inch-long log. Cut into 12 even rolls (about 1 1/2 inches in width each). Arrange them in the prepared baking dish, cut sides up. Cover the rolls very tightly with aluminum foil and place in the refrigerator for 8-12 hours. (See note below for making rolls the same day)
Remove rolls from the refrigerator and let rise in a warm place until they are doubled in size, about 1-2 hours.
Preheat the oven to 375°F. Bake for about 25 minutes, until they are golden brown. About halfway through the bake time, cover the rolls loosely with aluminum foil so the tops don't brown too much. Remove pan from the oven and allow to cool on a wire rack for about 10 minutes as you make the icing.
Make the icing: Whisk all of the icing ingredients together. If it's too thin, add a little more confectioners' sugar. Drizzle icing over warm rolls. Cover leftover frosted or unfrosted rolls tightly and store at room temperature for up to 3 days.