Preheat the oven to 350 degrees F. Generously butter the sides of a 10-inch cast iron skillet.
Mix together the softened butter, ½ cup of brown sugar, and a pinch of kosher salt. Spread the mixture evenly on the bottom of the skillet.
In a large mixing bowl, whisk together the flour, baking powder, baking soda, cinnamon, cardamom, nutmeg and ¾ teaspoon kosher salt.
In another bowl, whisk together the melted butter, vanilla, brown sugar and granulated sugar. Then whisk the eggs into the sugar mixture one at a time.
Add the yogurt to the wet ingredients and whisk until smooth.
Place the apples in the skillet in an overlapping circular pattern with the peel side touching the butter/brown sugar mixture. (So it looks pretty when you flip it!)
Add the dry ingredients to the wet ingredients in two batches. Use a silicone spatula to fold them together, smoothing out any lumps of flour as you go. Transfer the batter to the skillet, smoothing it out evenly with an offset spatula or silicone spatula. (See note below.)
Bake on the center rack for 40-45 minutes, until it is a deep golden brown, it springs back to the touch, and a toothpick inserted into the center comes out clean.
As soon as the cake comes out of the oven, run a knife or offset spatula around the outside edge. Let it cool in the skillet for 10 minutes. Then carefully flip it out onto a plate and let cool for at least 30 minutes before serving.