Preheat the oven to 350°F.
Grease a 9×5-inch loaf pan or line it with parchment paper.
In a medium bowl, mix the flour, baking soda, salt, and cinnamon.
In a large bowl, mix the brown sugar, granulated sugar, eggs, applesauce, melted butter, and vanilla until smooth.
Add the dry ingredients to the wet ingredients and stir just until they are combined. You don't want to overmix.
Gently fold in the chopped apples.
Pour the batter into the prepared loaf pan and smooth the top.
Bake for 50–60 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.
Let the bread cool in the pan for about 10 minutes, then transfer it to a wire rack to finish cooling.