Sift together flour, baking powder, soda, salt, and nutmeg. Beat shortening until creamy. Add sugar gradually and continue beating until light and fluffy. Add eggs and beat well. Add bananas, milk, and vanilla to sugar mixture and blend. Add dry ingredients and mix until smooth. Turn a small amount of dough onto a floured board. Knead very lightly. Roll out 1/2 inch thick. Cut with floured 2 1/2 inch doughnut cutter.
Heat fat or salad oil in a heavy kettle or deep fat fryer. Have fat 3 to 4 inches deep. Heat to 375 on a deep-fat thermometer or until a 1-inch cube of bread browns in 40 seconds. Slip doughnuts into fat with a spatula. Fry about 3 minutes, or until golden brown, turning frequently. Drain on absorbent paper. If desired, sprinkle with sugar or powdered sugar or a mixture of sugar and cinnamon.
Yield: 3 1/2 dozen doughnuts