Preheat oven to 350°F (175°C). Grease a 9x5-inch loaf pan, line with parchment paper, and set aside.
Peel, core, and dice apples into 1/2-inch cubes. Toss with 3 tablespoons sugar and 1 teaspoon cinnamon, then set aside.
In a small bowl, stir together brown sugar and 1 teaspoon cinnamon. Add melted butter and mix until combined.
Cream butter and granulated sugar until pale and fluffy. Beat in eggs one at a time, then add vanilla and sour cream.
Whisk together flour, baking powder, baking soda, salt, and 1 teaspoon cinnamon. Add to wet mixture and mix just until combined.
Spoon half the batter into the pan, top with half the apples, and drizzle with half the cinnamon sugar mixture. Repeat with remaining batter, apples, and topping.
Bake at 350°F for 55-65 minutes until a toothpick inserted in the center comes out clean. Cool in pan for 15 minutes, then transfer to a rack to cool completely.
Whisk powdered sugar, vanilla, and 2 tablespoons milk until smooth. Add more milk if needed. Drizzle over cooled bread and let set before slicing.