Preheat the oven to 350°F. Lightly grease and flour a 9-inch cake pan. Place the butter and sugar in a bowl of a mixer fitted with a paddle and mix until creamy. Add the molasses and mix until creamy. Add the sour cream and mix until creamy. Add the eggs, one at a time, beating well after each addition. Add the flour, baking powder, baking soda, salt, ground ginger and ginger root and mix just to combine. Place the batter in the prepared pan and smooth down. Place the apples on top in a circle, shingle style, around the edges of the cake. Transfer the cake to the oven and bake until the center is set and springs back under pressure, about 30 minutes, basting every 10 minutes with the melted butter. Serve warm or at room temperature. Cover and store at room temperature up to 2 days.
Place all the ingredients in a large pot and cook, covered, over medium heat for 20 minutes. Stir and continue to cook for 20 to 30 minutes. Repeat one to two times or until the apples are falling-apart soft, about 40 to 60 minutes. Remove and discard the cinnamon sticks. Transfer the mixture to a food processor fitted with a steel blade and purée. If you didn't peel the skins, you will now need to strain the mixture through a sieve. Cover and refrigerate up to 1 week.
If necessary, heat the applesauce. Slice the Apple-Topped Gingerbread into 10 pieces. Pour hot applesauce on top.