Butter 3 (9-inch) cake pans with 1 tablespoon of the butter and set aside. In a large bowl, cream the butter with an electric mixer. Add the sugar, and beat. In a medium bowl or on a piece of parchment, combine the flour, baking soda, cinnamon, and salt, and mix well.
Add the dry ingredients, alternating with the eggs, beating well after the addition of each. Add the vanilla extract and mix. Add the carrots and beat on medium speed until well incorporated, about 2 minutes. Fold in the nuts. Divide between the 3 cake pans and bake until set and a cake tester inserted into the middle comes out clean, 25 to 30 minutes. Remove from the oven and let rest in the cake pans for 10 minutes. Invert onto wire racks, remove from the pans, and let cool.
For the Frosting: In a large bowl, cream together the cream cheese and butter until light and fluffy. Add the sugar gradually, beating constantly. Add the vanilla and pecans.
When the cake is cool, place 1 cake layer on a cake plate or stand. Spread the top with cream cheese frosting and top with a second and third cake layer, spreading the icing between each layer. Spread the icing around the sides of the cake and let harden slightly before serving. To serve, cut into wedges.