Preheat oven to 350ºF. Prepare a 10-inch Bundt pan with nonstick spray or oil.
In a medium bowl, whisk together flour, cocoa powder, baking powder, baking soda, cinnamon, ginger, allspice, nutmeg, and sea salt. Set aside. In a large bowl, cream the butter, sugar and brown sugar with an electric mixer until light and fluffy. Add eggs one at a time, beating well after each addition. Set aside. In a small bowl, combine applesauce, buttermilk, and vanilla. Set aside.
With mixer on low speed, blend dry ingredients to the butter mixture in three additions, alternating with applesauce mixture. Beating until just combined. Do not overmix. Pour and spread batter evenly into prepared Bundt pan. Bake in a 350°F oven for 30 to 35 minutes or until a toothpick inserted into the center comes out clean. Keep cake in the pan for 5 minutes and then invert onto a wire rack. Remove pan and cool completely.
While cake is cooling, prepare frosting by melting butter in a medium saucepan over medium heat. Add brown sugar, stirring until dissolved, about 2 to 3 minutes. Add cream and whisk until smooth. Set pan aside to cool for 10 minutes, stirring occasionally. Then, add powdered sugar, salt, and vanilla, whisking until the icing is smooth and creamy. Immediately drizzle the icing over the top of the cooled cake, allowing it to drip down sides and into center.
Store the cake in an airtight container at room temperature for up to four days.