Preheat the oven to 350° Fahrenheit.
Grease a cast iron skillet with butter and sprinkle 1 tablespoon of flour evenly over the butter.
Mix the flour, brown sugar, salt, and butter in a small bowl using a pastry cutter or fork. Keep mixing until the pieces look like small peas. Put the bowl in the fridge until you're ready to use it.
Combine the apples, brown sugar, and lemon juice in a medium bowl. Set it aside.
Whisk together the flour, baking powder, salt, cinnamon and baking soda in a separate bowl. Set aside.
Cream the butter in the bowl of your stand mixer on high using the paddle attachment until smooth, about 2 minutes. Add the granulated and brown sugar and continue mixing on high until smooth, about 3 minutes.
Add the eggs one at a time, mixing thoroughly before adding the second one. Add vanilla extract.
Add ⅓ of the flour mixture to the stand mixer bowl and beat on low increasing speed as you go along until combined. Add ½ of the sour cream and mix until combined. Continue adding the flour mixture and sour cream until you have a thick batter.
Spoon the batter into a prepared skillet and smooth it out.
Strain any juice off the sitting apples and scoop the apple pieces over the top of the batter in an even layer. Top the apples with an even layer of the crumb topping.
Bake for 50-55 minutes. The top will be very crispy and toasted brown and the middle of the cake will be puffed up a little higher than the edges.
Cool in the pan for at least 30 minutes before slicing and serving. Drizzle with caramel sauce when ready to serve.