Preheat the oven to 350 degrees F (180 degrees C). Generously butter an 8x4-inch loaf pan, and line with parchment.
Core the apples, and cut into approximately 1/4-inch cubes. The apples do not need to be peeled. Transfer into a mixing bowl and add brown sugar, cinnamon, and ginger. Toss with a spatula until thoroughly combined. Let stand for 5 to 10 minutes to macerate.
Add flour, salt, and baking soda to a small bowl and whisk until evenly combined. Set aside.
Combine egg, white sugar, vegetable oil, sour cream, and vanilla in a bowl. Whisk until everything is thoroughly combined, about 1 minute. Use a spatula to toss the apple mixture, and then transfer about 75% of it into the egg-sugar mixture. Stir to combine evenly. Add flour mixture; stir gently with a spatula just until flour disappears.
Transfer the batter into the prepared loaf pan. Tap the pan a few times on the counter to settle the batter. Spoon remaining apple mixture evenly over the top.
Bake in the upper center of the preheated oven until a bamboo skewer comes out clean, 55 to 60 minutes. If using an instant-read thermometer, the internal temperature at the center should read 200 degrees F (95 degrees C).
Remove from the oven, and let cool for 15 to 20 minutes. Remove from the pan to a wire rack to cool completely, about 30 minutes.
Meanwhile, for glaze, whisk confectioner’s sugar, vanilla, and milk together until smooth; set aside.
Drizzle icing over cooled bread. Let the loaf sit until icing firms up. Serve immediately, or refrigerate to serve later.