Start by heating your oven to 350°F and having a 9x13 dish set aside. Then whisk the eggs, pumpkin, sugar, oil, and 1 tsp. vanilla extract together until combined and smooth. In a separate bowl, sift the flour, xanthan gum, baking powder, baking soda, salt, cinnamon, and nutmeg together. Then add the dry ingredients to the wet ingredients and stir until combined.
Pour the batter into your ungreased baking dish and bake for 30-35 minutes or until a toothpick inserted into the middle comes out clean. Allow the cake to cool completely.
While the cake is cooling, whisk together the butter and vanilla until smooth before adding in the powdered sugar. Beat the mixture on high until smooth and fluffy. Then spread the frosting over the top of the cake and serve!