Sift flour, measure and resift 3 times with baking powder, soda, salt, and nutmeg. Cream shortening, blend in sugar, add vanilla, and eggs, and beat until light and fluffy. Add combined bananas and buttermilk, and stir until well mixed.
Add flour mixture in 3 or 4 portions, stirring thoroughly after each addition. Chill before rolling. Remove 1/4 of dough from refrigerator at a time, knead it lightly 4 or 5 times, roll to 3/8 inch thickness, and cut with floured 2-1/2 inch doughnut cutter.
Fry in deep oil heated to 375 deg F. until golden brown, then lift out and drain on absorbent paper. Dough may also be covered tightly and kept in the refrigerator for 1 to 2 days, to be fried as needed.