Preheat the oven to 350°F (175°C). Prepare an 8-inch (20 cm) square or round baking tin by greasing it thoroughly and dusting it lightly with flour.

In a medium-sized bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg until uniformly combined.

In a separate large mixing bowl, beat the softened butter with both the granulated and brown sugars using a hand mixer or stand mixer until the mixture becomes light in color and airy in texture—this should take approximately 2 minutes.

Incorporate the eggs one at a time, ensuring each is fully mixed in before adding the next. Stir in the vanilla extract.

Add half of the dry mixture into the creamed mixture and mix just until combined. Then fold in the sour cream, followed by the remaining dry mixture, mixing only until no streaks of flour remain.

Using a spatula, gently stir in the diced apple pieces, distributing them evenly throughout the batter.

Transfer the batter into the prepared cake tin, smoothing the surface evenly.

In a small bowl, mix together the brown sugar, cinnamon, and chopped nuts, if using. Evenly scatter this mixture over the top of the batter.

Place the cake in the preheated oven and bake for 40 to 45 minutes, or until a toothpick inserted in the center emerges clean.

Allow the cake to cool in the pan for 10 minutes, then carefully remove and transfer it to a wire rack to cool further. Serve warm or at ambient temperature.