Preheat the oven to 350°F (175°C). Lightly grease a loaf pan.
In a bowl, combine the chopped apples, brown sugar, and cinnamon. Set aside.
In a large mixing bowl, whisk together the egg, granulated sugar, vegetable oil, sour cream, and vanilla extract until well combined. Add the flour, baking soda, and salt. Mix everything together until completely incorporated.
Fold 3/4 of the apple-cinnamon mixture into the batter. Pour it into your prepared loaf pan. Spread the remaining 1/4 of the mixture over the top.
Bake for 50–55 minutes, or until a toothpick comes out clean.
Cool the bread in the pan for about 10 minutes, then turn it out onto a wire rack to cool completely.
For the glaze, whisk together the powdered sugar, vanilla extract, and milk until smooth.
Pour the glaze over the bread before serving.