Preheat the oven to 350°F. Line the bottoms of 3 greased and floured 9-inch round cake pans with parchment paper.
Stir together the carrots, 2 cups of the flour, sugar, baking soda, baking powder, cinnamon, and salt; set aside.
Beat the eggs in a large bowl with an electric mixer at medium speed until smooth and thick, about 2 minutes.
Add the oil and vanilla to the eggs; beat until combined.
Gradually add the flour mixture to the egg mixture, beating well.
Divide the batter evenly among the pans.
Bake until a wooden pick inserted in the center comes out clean, about 30 minutes.
Cool in the pans for 15 minutes; remove from the pans, and cool completely on wire racks, about 30 minutes.
Make the Cream Cheese Frosting
Beat the powdered sugar, cream cheese, and butter with an electric mixer at medium speed until well blended; stir in the pecans.
Spread the frosting between the layers and on the top of the cake.
Top with the chopped pecans and the carrot ribbons, if desired.
Notes
To make carrot ribbons, peel the carrots. Holding the carrot by the thicker end, slide the vegetable peeler away from you for thin ribbons. Continue to shave from the same place to get wide ribbons.