Preheat oven to 350 degrees F (175 degrees C). Grease and line a 9x5 inch loaf pan with parchment paper.
In a medium bowl, whisk together the flour, granulated sugar, baking powder, baking soda, cinnamon, and salt. Set aside.
In a large bowl, stir together the oil, applesauce, yogurt, eggs, and vanilla extract until well blended.
Stir the dry ingredients into the wet ingredients until just combined. Gently fold in the diced apples.
Pour the batter into the prepared loaf pan. Sprinkle the brown sugar and walnuts (if using) over the top.
Bake in the preheated oven for 45-50 minutes, or until a toothpick inserted into the center of the loaf comes out clean.