Preheat oven to 350°F. Spray an 9x5-inch loaf pan with nonstick spray, grease with butter, or line with parchment paper.
In a small bowl, toss together the diced apples, granulated sugar, and cinnamon until the apples are well coated. Set aside.
In another small bowl, stir together the dark brown sugar and cinnamon until combined. Set aside.
In a large bowl, whisk together the egg and granulated sugar until smooth and well combined.
Add the melted butter, sour cream, and vanilla, and whisk until smooth.
Add self-rising flour, baking soda, and cinnamon and stir until just combined.
Pour half the batter into your prepared pan and gently smooth the top with a spatula. Evenly scatter half the apple mixture in a flat layer, then sprinkle half the brown sugar layer on top. Pour the remaining batter over and smooth it all the way to the edges of the pan. Top with the remaining apples and remaining brown sugar mixture.
Bake for 38 to 42 minutes until the top is domed and set and a toothpick inserted in the center comes out clean or with just a few moist crumbs. In the last 10 minutes, loosely tent the pan with a sheet of foil to keep the top and sides from getting too dark before the center is done.
Cool in the pan on a wire rack for at least 30 minutes
In a small bowl, whisk together powdered sugar and milk until smooth and pourable. If you'd like a thinner glaze, add a little more milk, one teaspoon at a time.
Drizzle the glaze evenly over the cooled bread, letting it run down the sides. Allow it to set for a few minutes before slicing.