Cream butter and sugar. Add beaten egg and mix well. Add vanilla.
In a separate bowl combine flour, baking soda, salt, nutmeg and cinnamon. Add to creamed mixture. Add grated apples.
Pour mixture into a greased 9 x 9 inch baking pan. Bake in a 350 degree oven for 30 minutes.
For frosting: Bring brown sugar, butter and milk to a boil. Boil only 2 1/2 minutes. Remove from heat and add sifted powdered sugar while stirring. Cool slightly and spread on bars. Sprinkle chopped pecans on top.
Cool completely and cut into bars. Store in air-tight container to retain moisture.
Note: If you prefer sweeter, add 1/4 cup sugar to batter of bars, but remember the apple juice from the grated apples will provide some sweetness. We enjoy it as is because the carmel frosting is sweet. Also, we have had them with no frosting and just a sprinkling of powdered sugar on top. Enjoy!