4 Tablespoons Unsalted Butter, Plus Extra For Greasing Cake Pan
1/2 Cup Packed Light Brown Sugar
1/8 Teaspoon Table Salt
4 Granny Smith Apples, Peeled, Cored, And Cut Into 1/2-Inch Slices (About 5 Cups)
1/2 Cup Sour Cream
1 Large Egg Plus 1 Yolk
1/2 Teaspoon Vanilla Extract
1 1/4 Cups All-Purpose Flour
3/4 Cup Granulated Sugar
1/2 Teaspoon Baking Powder
1/4 Teaspoon Baking Soda
1/4 Teaspoon Table Salt
8 Tablespoons Butter (1-Stick), Cut Into Chunks, Room Temp
Adjust oven rack to center position and heat oven to 350 degrees. Lightly butter 9-inch cake pan.
For the Apples: Place butter in large heavy-bottomed skillet over medium-high heat. Once butter has stopped foaming, add sugar and stir to combine. Continue to cook until sugar turns dark brown (it should look like dark brown sugar), about 2 minutes, swirling pan occasionally. Add salt and apples and fold with spatula to combine. Cook, stirring often, until apples have softened slightly and juices are thickened and syrupy, 5 to 7 minutes. Remove pan from heat and spoon apples into prepared cake pan.
Cake: In small bowl, whisk together 1/4 cup sour cream, egg, yolk, and vanilla until well combined.
Place flour, sugar, baking powder, baking soda, and salt in large bowl. Use electric mixer on low speed for 15 seconds to blend. Add butter and remaining 1/4 cup sour cream and mix on low speed until dry ingredients are moistened, 1 to 2 minutes. Increase speed to medium and mix for 2 minutes. Add sour cream/egg mixture and beat on medium-high, scraping down sides of bowl, until batter is homogeneous and fluffy, about 1 minute.
Spoon batter over apples and gently spread out to thin layer that covers apples. Bake until cake is dark golden brown, feels set, and cake tester comes out clean when inserted in center, 35 to 40 minutes. Transfer pan to wire rack and let cool for 5 minutes.
Place serving plate over top of pan and invert. Let cake sit inverted for about 1 minute without tapping or shaking pan. Cake will slowly detach itself. Once cake is on platter, gently remove pan. Serve warm or at room temperature with whipped cream or vanilla ice cream