Preheat the oven to 375 degrees . Butter and flour a 9- by 13-inch baking pan .
In a large bowl , combine the dry ingredients by whisking together the flour , baking powder , baking soda , salt , ground ginger , nutmeg , and cinnamon .
In a stand mixer with the paddle attachment , first beat the brown sugar on low until it is all loosened up . Add the vanilla extract . Then gradually beat in the eggs one at a time , until smooth and incorporated . Scrape down the sides and add the oil in a slow steady stream . Finally , mix in the applesauce .
Use a rubber spatula to fold the dry ingredients in three batches . Do not overmix . Fold in the carrots .
Pour the batter into the baking pan , smoothing out the top . Bake until the cake has risen and a cake tester inserted into the middle comes out clean , about 35-40 minutes . Transfer to a wire rack . Allow to cool completely .
In the meantime , prepare the ginger sherry syrup , candied pecans , and mascarpone cheese frosting .
For the ginger sherry syrup :
Bring sugar , water , and sherry to a simmer in a saucepan , stirring until sugar has dissolved . Turn of the stove . Add the grated ginger . Let cool completely .
For the candied pecans :
Brown the butter over medium-high heat . Add pecans and toss to coat . Add sugar and salt and stir until caramelized . Spread on wax paper and cool .
For the mascarpone cheese frosting :
Whisk together the first 4 ingredients in a large bowl just until blended . Beat whipping cream at medium speed with an electric mixer until stiff peaks form . Gently fold into mascarpone mixture .
Once the cake and syrup are cooled , use a cake tester to poke small holes all over the cake . Brush all over with the syrup . Tear cake into bite-sized pieces .
To assemble :
In 8 12-ounce glasses , add about 1 inch high of cake pieces , then 2 tablespoons of frosting , and then 1 single layer of candied pecans . Repeat the first two layers . Top with additional chopped candied pecans . Refrigerate for up to 24 hours if not eating immediately .