Grease and lightly flour two 8x1-1/2-inch or 9x1-1/2-inch round baking pans or grease one 13x9x2-inch baking pan; set pan(s) aside.
In a large mixing bowl stir together flour, sugar, baking powder, baking soda, and salt. Add bananas, buttermilk or sour milk, shortening, and vanilla. Beat with an electric mixer on low speed until combined. Add eggs; beat on medium speed for 2 minutes. Pour batter into the prepared pan(s).
Bake in a 350 degree F oven for 25 to 30 minutes for round pans or about 35 minutes for 13x9-inch pan or till a wooden toothpick comes out clean. Cool layer cakes in pans on wire racks for 10 minutes. Remove layer cakes from pans. Cool thoroughly on racks. Or, place 13x9-inch cake in pan on a wire rack; cool thoroughly. Frost with desired frosting. Makes 12 to 16 servings.