Preheat oven to 350°F.
Add the cream cheese, butter and sugar to the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed until well combined and smooth. Add the egg, flour and vanilla and continue beating just until incorporated. Set aside.
Place the pecans on a rimmed baking sheet and toaste in the oven for 8-10 minutes, or just until fragrant. Remove the pan to a wire rack.
In a large bowl, whisk the flour, both sugars, the cinnamon, salt, baking soda, nutmeg and allspice together. Add the eggs, oil, applesauce and vanilla and mix just until combined (I started with a whisk and switched to a rubber spatula). Fold in the toasted pecans and chopped apples until distributed throughout the batter.
Spoon about 1/2 to 2/3 of the cake batter into the prepared 12-cup bundt pan. Top evenly with the cream cheese filling, leaving a 1-inch border around the edge of the pan (I found this tricky, just do the best you can). Use a thin paring knife to swirl the cream cheese filling with the cake batter just a few times - less is more. Top the filling with the remaining cake batter.
Bake for 60-75 minutes, or until a thin knife inserted in the center of the cake comes out clean. Transfer the pan to a wire rack and cool for 15 minutes, then invert the cake onto the rack and allow to cool completely (at least 2 hours).
To make the frosting: Combine the brown sugar, butter and milk in a 2-qt saucepan. Set over medium heat, and bring to a boil, whisking almost constantly. Boil for 1 minute, whisking constantly. Remove the pan from the heat and stir in the vanilla. Whisk in the confectioners' sugar, a little at a time, until it is completely incorporated and the frosting is smooth. Gently stir the frosting until it starts to thicken, then pour it over the cooled cake. (The frosting will set up quickly, so don't make it in advance - wait until the cake has cooled completely.) Garnish with extra pecans, if desired.