reheat your oven to 350°F and butter and flour a 9X13in pan. I like to place a round of parchment on the bottom of the pan buttered and floured.
Sift together all dry ingredients and set them aside.
In the bowl of a stand mixer, or with a hand mixer in a large bowl, whisk the cold eggs on medium-high speed until they become light in color and pile on top momentary (about 5 seconds) when you lift the beaters out of the bowl, this should take about five minutes. Once the eggs are light in color and ready, gradually add the sugar a little at a time until fully incorporated. Next, reduce the speed of the mixer to medium-low and add your cooled melted butter, buttermilk, and vanilla. Once incorporated on low speed or by hand with a silicone spatula, add your sifted dry ingredients, being careful not to over mix. If using a stand mixer, make sure to give a few turns by hand to avoid dry ingredients that may be collected on the bottom of the bowl. Next mix in carrots and freshly grated ginger and crystallized ginger.
Pour into the prepared baking pan and bake in the preheated oven for about 50 minutes. You can test the cake by inserting a toothpick into the center. The cake is finished when it comes out clean. Once the cake is made, allow it to cool for about 15 minutes in the pan and then turn out onto a cooling rack to finish cooling.
While the cake is cooling, with a hand mixer and a large bowl whisk together the butter and cream cheese until fully incorporated. Add the vanilla bean paste. Then a little at a time add the powdered sugar until thoroughly combined. Make sure to scrape down the sides of the bowl and along the bottom at least once to avoid dry spots.
Once the cake is completely cool, you can cut it in half to make a layered cake or frost it as a whole cake. You will have plenty of frosting to do whichever appeals to you. Once frosted, add the crystallized ginger and shredded carrots to the top, and if you have one, the frosted center layer.
Enjoy this cake within two days of making it and store covered in the fridge.