In the bowl of a stand mixer, combine the warm milk, 1 tablespoon of granulated sugar, and the active dry yeast. Let the mixture sit for about 5 to 10 minutes until it becomes foamy.
Add the remaining granulated sugar, melted butter, and eggs to the yeast mixture. Whisk until well combined.
Add 4 cups of all-purpose flour and the salt. Using the dough hook attachment, mix on low speed until the flour is incorporated. Gradually add the remaining 1/2 cup of flour if the dough is too sticky.
Increase the mixer speed to medium and knead the dough for 5 to 7 minutes, or until the dough is smooth, elastic, and clears the sides of the bowl.
Place the dough in a lightly greased bowl, cover with plastic wrap or a clean kitchen towel, and let it rise in a warm, draft-free spot for 1.5 hours, or until it has doubled in size.
While the dough rises, prepare the filling by whisking together the brown sugar, granulated sugar, and ground cinnamon in a small bowl. Toss the diced apples with 1 tablespoon of the cinnamon-sugar mixture.
Punch down the risen dough and turn it out onto a lightly floured surface. Roll the dough into a large rectangle, approximately 12x18 inches.
Spread the 4 tablespoons of softened butter evenly over the dough, leaving a 1/2-inch border at the edges. Sprinkle the cinnamon-sugar mixture over the butter, then distribute the diced apples evenly over the top.
Starting from the long edge, roll the dough up tightly into a log. Use a sharp serrated knife or unflavored dental floss to cut the log into 12 even rolls.
Place the rolls into a greased 9x13 inch baking pan. Cover and let them rise again for 45 minutes.
Preheat your oven to 375°F (190°C). Bake the rolls for 22 to 25 minutes, or until they are lightly golden brown on top.
While the rolls bake, prepare the icing by beating the cream cheese and softened butter until smooth. Gradually add the powdered sugar, vanilla extract, and milk, whisking until creamy.
Allow the rolls to cool for 5 to 10 minutes before spreading the cream cheese icing generously over the top. Serve warm.