Make the filling. Place cream cheese, sugar, and egg in a medium bowl. Use a hand-held mixer to beat until smooth. Cover with plastic wrap and refrigerate for 45 to 60 minutes.
Once cream cheese mixture has refrigerated long enough to thicken up, make the batter.First, preheat the oven to 350 degrees and spray a Bundt pan with baking spray with flour.Combine the carrots, granulated sugar, brown sugar, oil, and eggs in a large bowl. Beat until blended well.
In a medium bowl, whisk together flour, baking soda, cinnamon, and salt.
Gradually beat the flour mixture into the carrot mixture. Stir in nuts.
Bake for 55 to 60 minutes. Let cool in pan for 5 minutes and then invert onto a cake stand or serving platter. Let cool completely before frosting.
Beat cream cheese and butter with a hand-held mixer until smooth. Add vanilla extract and 1 cup of powdered sugar and beat until smooth.Gradually beat in remaining sugar. If too thick, add enough milk to thin it some.