In the bowl of the stand mixer, stir together the yeast, warm milk, and 1 teaspoon granulated sugar. Let sit for 5 minutes to foam.
Add the brown sugar, salt, melted butter, and mix on medium speed using the dough hook. Slowly add in the flour ½ cup at a time until a soft dough forms.
Knead for 10 minutes using your dough hook. Remove the dough to a large floured bowl, cover with a heavy towel and place in a warm spot to rest for 1 hour.
While the dough is rising, make the apple filling. In a large pan, melt 2 tablespoons of butter over low-medium heat. Add the diced apples and cook for 7-10 minutes, until softened. Toss with ½ cup of the brown sugar and 1 teaspoon cinnamon. Cook for 1-2 minutes more. Remove from the heat.
In a separate bowl, melt the remaining 4 tablespoons of butter. Add the remaining cup of brown sugar and 3 teaspoons cinnamon. Mix well.
Roll the risen dough out to a rectangle about 9 inches by 12 inches. Spread the brown sugar butter filling over the dough, leaving about a ½ inch edge on all sides. Spread the apple filling on top, again leaving a ½ inch edge.
Cut the dough into 12 equal strips. Roll and place in a well greased baking dish. Cover with a towel and let rise for 1 hour in a warm place.
Bake at 350℉ for 30 minutes. Remove from the oven and let cool slightly before icing.
While baking, make the icing by beating together the icing ingredients over low speed in a large mixing bowl. Spoon icing over the slightly cooled rolls and serve warm.
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