Heat the oven to 375 F. Grease and flour two deep 8-inch round cake pans.
Put the shortening in a large mixing bowl.
Sift the flour, baking powder, salt, sugar, and pumpkin pie spice together over the shortening.
Add ½ cup of the milk and the pumpkin. Beat with a mixer for 1 minute.
Add the remaining ¼ cup milk and the eggs. Beat 2 more minutes, scraping the bowl often.
Divide the batter between the pans. Bake about 35 minutes, or until a toothpick inserted in the center comes out clean.
Cool the layers completely before frosting.
Frosting directions
In the top of a double boiler, beat the egg whites, brown sugar, water, corn syrup, and salt for about 1 minute, until blended.
Set the pan over simmering water, without letting the bottom touch the water. Beat on high for about 7 minutes, until the frosting is glossy and holds stiff peaks. If you have an instant-read thermometer, it should reach at least 161 F.
Take the pan off the heat. Beat in the vanilla for 1 to 2 minutes, until thick and fluffy.
Spread it between the layers and over the top and sides right away, because it sets quickly.