Preheat the oven to 350°F. Grease or line a 9x5-inch loaf pan with parchment paper.
Add the diced apples, 2 tablespoons brown sugar, and 1 teaspoon cinnamon to a bowl. Stir to coat and set aside.
In a small bowl, mix together the ⅓ cup brown sugar and 1½ teaspoons cinnamon for the cinnamon sugar swirl.
In a large bowl, beat the softened butter and cane sugar until creamy.
Add the eggs one at a time, then mix in the vanilla, Greek yogurt, and milk.
In a separate bowl, whisk together the flour, baking powder, baking soda, and sea salt.
Add the dry ingredients to the wet ingredients and gently mix just until combined. Do not overmix.
Spread half of the batter into the prepared loaf pan. Top with half of the apple mixture and half of the cinnamon sugar.
Spread the remaining batter over the top, then add the remaining apples and cinnamon sugar.
Gently swirl a butter knife through the batter a few times.
Bake for 50–60 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs. Loosely cover with foil if the top begins to brown too quickly.
Let the bread cool in the pan for about 15 minutes before transferring it to a wire rack.
Whisk together the powdered sugar, milk, and vanilla until smooth. Drizzle over the cooled or slightly warm bread.