In a small bowl, add the yeast and 2 teaspoons of the sugar to the water. Allow it to sit and for about 5-10 minutes until foamy.
In the bowl of a stand mixer fitted with the dough hook attachment or in a large bowl if mixing by hand, mix together the milk, melted butter, egg, remaining sugar, and salt on low speed until well combined. Add the yeast mixture and continue to mix. Slowly add in the flour mixture until a dough forms. If needed, add more flour about 1/4 cup at a time. The dough should be pliable but not too sticky.
Knead the dough on a lightly floured surface for about 2-3 minutes. Then place it into a large well-oiled bowl. Cover and allow it to double in size, about 2 hours.
Meanwhile, make the filling. In a medium sized saucepan, melt 4 Tablespoons of the butter and sugar. Add the apples and saute until the apples are soft and slightly caramelized. Add 1 teaspoon of the cinnamon and coat well. If the apples are too soggy, drain before the next step. You do not want the apples to be too saucy - you can save the sauce to top the rolls.
Preheat the oven to 375 degrees F. Generously grease a 9"x13" pan and set aside.
Once the dough has doubled in size, take off the cover and punch the dough down with your fists. Transfer the dough to a lightly floured surface and roll it out to a 16"x24" rectangle.
In a small bowl, combine the brown sugar with the 2 Tablespoons of cinnamon. Mix with a fork and set aside.
Spread the remaining 2 Tablespoons of butter onto the dough rectangle. Then sprinkle the top with the brown sugar/cinnamon mixture and make sure it is spread out evenly. Top with the cooked apples, being careful not to add any of the liquid.
Roll the dough up, jelly-roll style, starting with the longest end of the rectangle. Pinch the seams down and cut the large roll into 12 even slices (a bench scraper works great here).
Place each roll, cut side up, on your prepared baking pan. If desired, you can top with the reserved apple liquid. At this point, you can cover and store in the refrigerator and bake the next day (allow to thaw for about 20 minutes prior to baking), or you can cover it and allow it to rise another time, for about 1.5-2 hours.
Preheat your oven to 375 degrees F.
Once the rolls have finished rising, place it into your preheated oven for about 20-22 minutes or until the tops start to turn golden in color. If desired, you can add a glaze or cream cheese icing on top. I chose to keep mine plain since they were already sweet enough.