Place a rack in the center of your oven. Preheat oven to 350 degrees Fahrenheit 20 minutes before baking your bundt cake. Grease and flour your bunt pan and set aside.
For the cream cheese filling, mix cream cheese and sugar in a bowl until smooth. Add your egg to the mixture and stir until incorporated.
In a large bowl, whisk sugar, oil, and eggs until combined. In a medium bowl, stir flour, baking powder, cinnamon, salt, and baking soda. Slowly stir the flour mixture into the large bowl with sugar. Finally, incorporate apples, carrots, and pecans. Take care not to over-mix the batter at this point.
Pour half of your cake batter into the pan. Add a layer of your cream cheese mixture. Then top with the remaining cake batter. Tap your pan a few times on the counter to remove any air bubbles and then smooth the top.
Bake your cake for 50-60 minutes. To check for doneness, gently press down on the cake. If it bounces back completely, your cake is done! If your finger leaves an indent in the cake, it needs more time. Cool for 10 minutes before inverting onto a wire rack to cool completely.
While the cake is cooling, make your icing. Bring brown sugar, butter, and milk to a boil in a medium saucepan. Stir continuously for one minute. Remove the pan and immediately stir in the confectioner's sugar and vanilla until combined. Drizzle the cooled mixture over the cake and sprinkle with pecans.