For the Rolls:

1. Preheat oven to 200 degrees F.
2. Meanwhile, combine the milk, yeast and 2 tablespoons of the sugar together in a bowl.  Whisk to combine and set aside for 10 minutes (the mixture should bubble & froth a bit – this indicates the yeast is good)
3. In your stand mixer bowl (fitted with the dough hook), add yeast mixture, add egg, flour and remaining sugar.  Beat on high for 2 minutes or until dough is smooth & elastic.  Alternatively, mix and knead by hand on a lightly floured surface, adding small amounts of flour from the surface as needed if it is too sticky to work with).
4. Turn oven off.  Lightly coat a large bowl with oil or cooking spray and transfer the dough to the bowl, turning to coat in oil. Cover with a damp towel.
5. Place in the now off oven to rise for 30 minutes. (dough has risen fully when you press your fingers gently into it and the indentation recedes.
6. After 30 minutes, punch dough down, then re-cover and let rise for an additional 30 minutes or until doubled in size.
7. On a lightly floured surface, roll out dough into a large rectangle.
8. Brush the melted butter over the dough.
9. Combine brown sugar, cinnamon and apple pie spice.  Sprinkle half evenly over the butter, gently pressing into the butter.  Layer diced apples evenly over the cinnamon sugar mixture.  Follow with remaining cinnamon sugar mixture.
10. Beginning at one long side, roll up dough tightly like a jelly-roll; then gently pinch the seam down to seal. Trim the ends for even slices, and cut dough into 2-inch slices.  Place each slice into a greased 9×13 inch pan.
11. Bake for 30-40 minutes or until lightly golden brown.

For the Icing:

1. Combine powdered sugar and milk stirring well.  Add additional milk or sugar to reach desired consistency.  Drizzle over rolls.