Preheat oven to 350 degrees F (180 degrees C).
In a large bowl, stir together All-Purpose Flour (2 cup), Granulated Sugar (2 cup), Baking Powder (2 teaspoon), Ground Cinnamon (1 teaspoon), Salt (1/2 teaspoon), and Baking Soda (1/2 teaspoon).
Stir in the Butter (1 cup).
Whisk Egg (4) and Vanilla Extract (1 tablespoon) then pour them into the dry ingredients and mix well to combine. Add the Carrot (3 cup) and gently fold them into the batter.
Pour the batter into a 9x13 inch pan and spread it evenly. Bake for 25 minutes or until a toothpick comes out clean.
Remove from the oven and let cool in the pan. While cooling, make the vanilla bean cream cheese by beating Cream Cheese (16 ounce) and Vanilla Bean Paste (2 teaspoon) with an electric mixer until smooth and creamy.
Turn the mixer on low and gradually add the Powdered Confectioners Sugar (2 cup) until the frosting reaches the consistency you prefer.
Spread the frosting on the cooled carrot bars. Top with Chopped Nuts (1 cup). Serve cold.