Preheat the oven to 350 ℉. Grease an 8x4 loaf pan with oil or butter and coat with flour.
In a large bowl whisk together the flour, sugar, cinnamon, baking soda, baking powder, and salt until well combined. In a separate bowl beat the eggs, oil, and vanilla together; add to the flour mixture stirring until just mixed. The dough will be very thick. Fold in the chopped apples and 1/2 cup of the walnuts. Pour the batter into the prepared pan.
In a small bowl toss the remaining walnuts with the brown sugar and sprinkle on top of the batter. Bake until a toothpick inserted into the center comes out clean; about 50 to 55 minutes.
Let cool in the pan for 10 minutes before removing to a wire rack to cool completely.