In a medium skillet, melt 1 tablespoon butter over medium heat. Add apples, apple pie spice, and 1 tablespoon brown sugar. Saute apples stirring occasionally until apples are a deep golden brown and tender, about 5-7 minutes. Allow to cool.
In a small bowl, combine remaining 3 tablespoons softened butter, remaining 1/3 cup sugar and cinnamon. Set aside.
Preheat oven to 350F. Lightly grease two 12-cup muffin tins and set aside. Punch risen dough down and turn onto a lightly floured surface. Roll dough out into a 14x8 inch rectangle.
Spread cinnamon sugar butter to within 1 inch of dough edges. Spread caramelized apples evenly over butter.
Tightly roll up the dough pinching edges to seal. Trim edges off and cut dough into 1 inch slices (you should end up with 12-14 rolls) Place cinnamon rolls in prepared muffin cups. Let rise at room temperature 30 minutes until doubled in size.
Bake cinnamon rolls at 350F 20-25 minutes until a light golden brown. Let cool in pan 15 minutes before transferring to a wire cooling rack.