Cream together butter, sugar and vanilla; beat in eggs. Measure flour, salt, baking powder and nutmeg in another bowl. Add dry to butter mixture and gradually add milk. Mix thoroughly into a soft dough. Pat dough to 1/3-inch thickness on cutting board. Cut out doughnuts and deep fry in melted shortening or vegetable oil at 380°F for approximately 2 minutes per side, or until golden brown.