Heat oven to 350°F. Spray 13x9-inch pan with non-stick cooking spray. Set aside.
For the cake, in a small bowl whisk together flour, baking soda, cinnamon, and salt. Set aside.
In a large bowl, combine eggs, buttermilk, vegetable oil, vanilla extract, and sugar. Mix until smooth. Slowly add in dry ingredients and mix until just combined. Gently fold in carrots, coconut, pineapple, and pecans. Pour batter evenly into prepared pan and bake 35 to 40 minutes or until cake springs back when pressed.
While the cake is baking, prepare the glaze by adding sugar, buttermilk, butter, corn syrup, and baking soda to medium-sized saucepan over medium heat. Vigorously and continuously stir mixture to subside the rising foam, for about 8 minutes. When the mixture is a light amber color, remove from heat and stir in vanilla extract. Set aside until ready to use.
When the cake comes out of the oven, immediately poke holes in cake using the handle of a spatula. Pour the buttermilk glaze over the top and spread evenly. Let cake cool completely.
While the cake cools, make the frosting by adding the cream cheese and butter in large bowl and beat until fluffy. Add brown sugar, cinnamon and vanilla bean paste and mix until combined. Slowly add powdered sugar and mix until smooth. Spread frosting evenly over cake and serve.