Preheat your oven to 350°F (175°C). Grease a 9x5-inch loaf pan and line it with parchment paper, leaving an overhang on the sides to easily lift the bread out after baking.
In a medium skillet over medium heat, melt the butter until it begins to foam slightly. Add the diced apples, brown sugar, cinnamon, nutmeg, and salt. Cook for 8-10 minutes, stirring occasionally, until the apples have softened and the sugar has caramelized into a rich sauce that clings to the fruit.
While the apples are cooling slightly, whisk together the flour, baking powder, baking soda, and salt in a medium bowl.
In a large bowl, beat the granulated sugar, brown sugar, and eggs until light and fluffy, about 2 minutes with an electric mixer. Slowly add the oil, continuing to beat until well incorporated. Stir in the Greek yogurt, vanilla extract, and milk until the mixture is smooth and uniform.
Gently fold the dry ingredients into the wet ingredients with a rubber spatula, mixing just until no flour streaks remain.
Reserve about 1/4 cup of the caramelized apples for topping. Fold the remaining caramelized apples and their sauce into the batter.
Pour the batter into the prepared loaf pan. Arrange the reserved caramelized apples on top and drizzle with any remaining sauce. Bake for 55-60 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs but no wet batter. If the top is browning too quickly, loosely cover with aluminum foil after 35 minutes.
Allow the bread to cool in the pan for 15 minutes, then use the parchment paper overhang to lift it out onto a wire rack to cool completely.