Begin by combining yeast and warm milk. Allow to dissolve completely, about 5-10 minutes.
Add sugar, eggs, and vanilla to stand mixer bowl. (You can mix by hand or with a hand mixer as well.)
Add baking powder, salt, and two cups of flour to mixer and mix on low speed with a dough hook. Add 1/2 cup of flour and bits of grated butter at a time until the rest has been incorporated. Dough should be slightly sticky, barely pulling away from the edges of the bowl.
Roll dough into a ball on a well floured surface. Wetting your hands may help transfer the sticky dough. Place dough in a bowl, cover, and allow to rise in a warm place about 2 hours.
While dough is rising, peel and slice apples.
Dice apples and heat in a skillet on medium-low heat. Add butter. Once apples are cooked and chewy (takes about 15 minutes) add brown sugar and cinnamon. Remove from heat and allow to cool.
Next, make your brown sugar mixture. Combine melted butter, brown sugar, flour, and cinnamon in a bowl.
After apples have cooled add them to the brown sugar mixture.
After dough has risen roll it out to about 1/4 in thickness on a well floured surface. (Flour the rolling pin as well to keep dough from sticking.)
Spread the apple/brown sugar mixture across the surface of the rolled out dough.
Using a bench scraper or a pizza cutter cut the dough into approximately 1 ½ x 8 inch strips. SEE PICTURE.
Roll the strips and place in a greased 9x13 pan, allowing space in between rolls for rising.
Allow rolls to rise about one hour or until nearly doubled.
Bake at 350 ℉ for about 20-30 minutes.
Allow rolls to cool 15-20. While rolls are cooling make frosting by combining confectioner's sugar and milk with a hand mixer. Pour over cooled rolls.