Start by preheating your oven to 350°F so it's fully heated by the time your batter is ready. Grease your baking pan generously, making sure to coat every corner so nothing sticks later. I like to dust it lightly with flour too, just for extra insurance against sticking.
In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg until evenly combined. This step distributes the spices so you don't end up with a cinnamon-heavy bite in one spot. Set this bowl aside while you work on the wet ingredients.
In your large mixing bowl, beat the eggs, granulated sugar, and brown sugar together until the mixture looks pale and slightly fluffy. Add the oil, vanilla, and buttermilk, then mix again until everything is fully combined. This forms the moist base that makes the cake so tender.
Gradually add the dry ingredient mixture into the wet ingredients, stirring just until no streaks of flour remain. Overmixing at this stage can make your cake dense, so stop as soon as it looks combined. A few small lumps are completely fine here.
Gently fold your diced apples into the batter using a spatula, making sure they're evenly distributed throughout. Don't overwork the batter while doing this, or you'll lose some of that light texture. The apples should peek through the batter in every scoop.
Pour the batter into your prepared pan and smooth the top with your spatula. Bake for 55 to 60 minutes, or until a toothpick inserted in the center comes out clean or with just a few moist crumbs. Baking times can vary slightly depending on your pan shape and oven.
Let the cake cool in the pan for about 15 minutes, then transfer it to a wire rack to cool completely. Once cooled, whisk together the powdered sugar and milk, then drizzle it generously over the top. Let the glaze set for a few minutes before slicing and serving.