To make the cake: heat the oven to 350°F. Butter and flour a 12-cup Bundt pan. Sift together the flour, cinnamon, baking soda, ginger, salt, pepper, and allspice into a bowl. Using another bowl and a handheld electric mixer—or a stand mixer fitted with the paddle attachment—beat the eggs and sugars until light and fluffy, about 3 minutes. Add the applesauce, oil, and vanilla and beat until smooth.
Fold in the dry ingredients until just combined. Pour the batter into the Bundt pan and bake for about 45 minutes, until a cake tester inserted into the center comes out clean. Transfer the pan to a rack and let cool for 10 minutes. Invert the cake onto the rack and let it cool completely, about 2 hours, before you make the glaze.
Put a piece of aluminum foil or parchment paper under the rack to catch any drips. To make the icing: in a saucepan, combine the brown sugar, cream, butter, and salt and cook over medium heat. Bring to a full rolling boil, stirring constantly. Boil for 1 minute exactly and turn off the heat.
Let the glaze cool in the pan for a minute, then gradually whisk in the confectioners’ sugar until you have a thick but pourable glaze (you may not need all of the sugar). If the mixture seems too thick, add a splash of cream. Pour the glaze quickly over the cake, covering as much surface area as possible. Let the glaze set for at least 10 minutes before storing cake in an airtight container for up to 5 days.