Preheat the oven to 350 and spray a quarter sheet pan (9" x 13" x 1") with cooking spray.
In a large bowl, whisk together the flour, sugar and salt.
In a separate bowl, whisk together the melted butter, buttermilk, egg white, pumpkin, baking soda and pumpkin pie spice. Mix very well.
Pour the pumpkin mixture into the bowl with the flour, and stir until everything is combined.
Pour the batter into the prepared pan and bake for 16-18 minutes, or until a toothpick inserted comes out clean.
While the cake cools, make the frosting: in a small sauce pan, melt the butter over medium heat. You can brown the butter if you like. Whisk in the buttermilk and powdered sugar. Add the pumpkin spice last, and whisk very well to combine.
Pour the warm frosting over the cake, let rest for about 15 minutes (or longer) before slicing and serving.