In a small bowl, toss chopped apples with 2 tablespoons of sugar and 1/2 teaspoon cinnamon. Set aside to marinate.
Preheat oven to 350°F (175°C). Grease and flour a 9x5-inch loaf pan. In a small bowl, mix together the brown sugar and cinnamon for the swirl.
In a larger mixing bowl, cream together butter and granulated sugar until light and fluffy. Add eggs one at a time, then mix in vanilla.
In another bowl, whisk together flour and baking powder. Add dry ingredients to the butter mixture alternately with milk, beginning and ending with flour. Stir just until combined.
Pour half the batter into the prepared loaf pan. Layer half of the apple mixture over the batter, then sprinkle with half the cinnamon-sugar mix. Repeat the layers with the remaining batter, apples, and cinnamon-sugar.
Use a knife to gently swirl the batter a few times for a marbled effect.
Bake for 50–60 minutes, or until a toothpick inserted in the center comes out clean. Let the bread cool for 15 minutes, then remove from the pan and cool completely on a wire rack.
For the glaze, whisk together powdered sugar, vanilla, and milk until smooth. Drizzle over cooled bread and allow it to set before slicing.