Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan and line the bottom with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a large bowl, beat the softened butter, granulated sugar, and brown sugar until light and creamy.
Add the eggs one at a time, then mix in the vanilla extract.
Add the sour cream and milk and mix until combined.
Gradually fold the dry ingredients into the wet ingredients. Mix only until no large streaks of flour remain.
Gently fold in the diced apples, making sure they are evenly distributed throughout the batter.
Spread the batter evenly into the prepared pan. Bake for 40–50 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.
Let the cake cool in the pan for about 15 minutes before transferring it to a serving plate.
For the caramel sauce, melt the butter in a saucepan over medium heat. Add the brown sugar and heavy cream and stir until smooth.
Bring to a gentle simmer and cook for about 3–4 minutes, stirring frequently. Remove from the heat and stir in the vanilla and salt.
Allow the caramel to cool slightly so it thickens, then generously spoon or drizzle it over the cake.