In a large skillet, melt the butter over medium heat. Add the apples, sugar and cinnamon.
Cook, stirring occasionally, until the apples are tender, about 10 minutes. Transfer the apples to a bowl and let cool to room temperature.
Crumb Topping
In a medium bowl, mix the flour, brown sugar, sugar, cinnamon and salt.
Work in the butter with your fingertips until the mixture looks like wet sand. Press together to make 1/2-inch chunks.
Cake
Preheat the oven to 325ºF. Line a 9-inch cake pan with a removable bottom*, or a 9-inch springform pan, with parchment paper. Wrap the bottom of the cake pan with aluminum foil and place pan on a baking sheet.
In a medium bowl, whisk together the flour, baking powder and salt.
In a large bowl, cream the butter and sugar, with an electric mixer on medium speed, until light and fluffy.
Beat the eggs and vanilla into the butter mixture in three separate addition, making sure each addition is incorporated before adding the next.
In a small bowl, whisk the milk and sour cream together.
On low speed, beat the flour mixture into batter in three separate additions, alternating with milk mixture. Mix just until combined.
Transfer the batter to the prepared baking pan. Smooth the top with a spatula.
Drain any excess liquid from the apples. Top the cake batter evenly with apples, then the topping mixture.
Bake until a toothpick inserted into the center of the cake comes out clean, about 50-60 minutes. Cool the cake in pan for 10 minutes. Remove the cake from pan and transfer it to a cooling rack to cool completely.
Icing
In a medium bowl, whisk together the powdered sugar, melted butter, vanilla, salt and 2 tablespoons of cream. Add additional cream until desired consistency is reached. Drizzle over the cooled cake.