Preheat oven to 350 degrees and spray a bundt pan with nonstick cooking spray with flour.
In a large bowl, beat together the white sugar, 1 cup light brown sugar, and melted butter with an electric mixer at medium speed.
Add eggs one at a time, scraping down sides of bowl between additions.
Beat in the vanilla extract.
In a medium bowl, whisk together the flour, baking powder, apple pie spice, and salt. Gradually add to sugar/egg mixture, beating until combined.
In a medium bowl, toss together apples, remaining brown sugar, and cinnamon.
Spoon 1/3 of batter into prepared pan. (Batter will be thick and you may need to spread it with a rubber spatula or lift the pan and drop it on the counter to get the batter to spread.)
Spoon 1/2 apple mixture on top of batter and repeat layers, finishing off with a third layer of batter.
Bake for 50 to 60 minutes, or until firm to the touch on top. Let cool for 10 minutes in pan and then turn cake out onto a wire rack.
FOR THE CARAMEL DRIZZLE: Mix brown sugar, half-and-half, butter, and salt in a small, heavy-bottomed saucepan. Cook over medium-low heat, whisking continuously, until mixture starts to thicken nicely, about 5 to 8 minutes. Remove from heat and stir in vanilla extract.
FOR CREAM CHEESE Icing: Use an electric mixer to beat together the cream cheese and confectioners' sugar until smooth. Beat in milk and 1/4 cup of the caramel drizzle.
Spoon Cream Cheese Icing on cake so that it drips down the sides. Spoon Caramel Drizzle on top of Icing. Garnish with pecans.