Make the dough: Heat milk to about 95°F (35°C)– use microwave or stovetop. Pour the warm milk into the bowl of a stand electric mixer fitted with the dough hook attachment (OR you can use a handheld mixer OR no mixer, but a stand mixer is ideal). With a whisk, manually whisk in the sugar and yeast. Cover with a towel and let sit until the yeast is foamy, about 5-10 minutes. If the yeast does not dissolve and foam, start over with fresh active yeast. 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. *If you do not have a stand-mixer with a hook attachment, knead the dough by hand in this step.
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.
Grease the bottom and sides of a 9×13 inch baking dish. Turn the risen dough out onto a lightly floured work surface, punch it down and, using a rolling pin, roll into a 12×18 inch rectangle. Make sure the dough is smooth and evenly thick.
Fill the rolls: Spread the softened butter all over the dough. In a small bowl, toss the brown sugar and cinnamon together until combined and then sprinkle evenly over the dough. Top evenly with chopped apples. Tightly roll up the dough to form an 18-inch-long log. Cut into 12 equal rolls. Arrange them in the prepared baking pan. Cover the rolls very tightly with aluminum foil.
Allow rolls to rise in a warm environment again for about 45-60 minutes or until doubled in size.
Bake the rolls: Preheat the oven to 375°F (191°C). Bake rolls for about 25 minutes until they are golden brown. About halfway through the bake time, I like to 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 5 minutes.
Make the icing: Whisk all of the icing ingredients together and drizzle over warm rolls. Serve warm. Cover leftover frosted or unfrosted rolls tightly and store for up to 3 days at room temperature or up to 5 days in the refrigerator.