For the cake, mix sugar and shortening until creamy. Add egg yolks and bananas, beat thoroughly. Dissolve soda in sour milk. Sift flour, baking powder and salt. Mix alternately. Add vanilla. Beat egg whites until stiff and fold into batter. Pour into a buttered 9-inch round cake. Bake in a preheated 350 degree oven for 40 minutes.
For the frosting,cream the sugar, shortening and butter until light and fluffy. Cook vanilla pudding with milk until very thick. Cool and add to butter mixture. Mix with electric mixer at high speed for 15 minutes or until it is not grainy. Add 1 teaspoon vanilla. Ice the cake with the icing.