Preheat your oven to 350°F (175°C) and grease a 9-inch round cake pan.

In a medium saucepan, melt together 4 tablespoons of unsalted butter and 1 cup of brown sugar over medium heat, stirring constantly until smooth and bubbling, then pour into the prepared pan and arrange sliced apples on top.

In a large bowl, whisk together 1 ½ cups of all-purpose flour, 1 teaspoon of baking powder, ½ teaspoon of baking soda, ½ teaspoon of salt, 1 teaspoon of cinnamon, and ¼ teaspoon of nutmeg.

Cream together ½ cup of unsalted butter, ¾ cup of granulated sugar, and ¼ cup of brown sugar until light and fluffy, then mix in 2 eggs, 1 tablespoon of vanilla extract, and ½ cup of sour cream or plain yogurt.

Gradually combine the dry mixture with the wet mixture until just blended, then pour this batter over the apples in the cake pan.

Bake for 40-45 minutes, checking doneness with a toothpick.

Let the cake cool in the pan for 10-15 minutes, then invert onto a serving plate.

Serve warm, optionally with whipped cream or ice cream.