Line the bottom of a and 9X5-inch loaf pan with parchment paper (I always like to do this to prevent the bread from sticking to the bottom of the pan). Lightly grease the insides of the pan with cooking spray.
In a large bowl, whisk together oil, granulated sugar and brown sugar.
Add the eggs, whisking until smooth. Add the applesauce, vanilla, and Greek yogurt and mix until well-combined.
Add the flours, baking soda, baking powder, cinnamon, nutmeg and salt and stir just until combined--don't overmix; a few lumps are ok.
Pour the batter into the loaf pan and spread into an even layer.
Bake for 45-55 minutes (ovens will vary on exact time) until a toothpick inserted into the center comes out clean with just moist crumbs.
Remove from oven. Run a knife gently around the side of the pan and then remove bread onto a cooling rack.