Preheat oven to 350°F (175°C). Grease and flour a 9×5 inch loaf pan.
In a medium bowl, toss the chopped apples with 1/3 cup brown sugar, 1 1/2 tsp cinnamon, and 1/2 tsp nutmeg. Set aside.
In a separate large bowl, whisk together the flour, baking soda, salt, 1 tsp cinnamon, and 1/4 tsp nutmeg.
In another large bowl, cream together the softened butter and 1 1/2 cups granulated sugar until light and fluffy.
Beat in the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
In a small bowl, whisk together the applesauce and milk.
Add the dry ingredients to the wet ingredients alternately with the applesauce mixture, beginning and ending with the dry ingredients. Mix until just combined, being careful not to overmix.
Gently fold in the coated apple chunks.
Pour the batter into the prepared loaf pan and spread evenly.
Bake for 60-75 minutes, or until a wooden skewer inserted into the center comes out clean. If the top begins to brown too quickly, loosely tent with foil.
Let the bread cool in the pan for 10-15 minutes before inverting it onto a wire rack to cool completely.