Preheat oven to 375 degrees. Butter a 12-by-1-inch round pizza pan. Line bottom with parchment paper, and butter parchment; set aside. Put 1/2 cup butter, the brown sugar, brandy, and a pinch of salt into the bowl of an electric mixer fitted with the paddle attachment; mix on medium-high speed until pale and fluffy, 2 to 3 minutes. Spread mixture into prepared pan. Place cinnamon sticks on top. Layer with apple rings. Use your hands to press apples gently into mixture; set aside. Sift flour, baking soda, baking powder, 3/4 teaspoon salt, and the ground cinnamon into a medium bowl; set aside.
Put remaining 3/4 cup butter and the granulated sugar into the clean bowl of an electric mixer fitted with the clean paddle attachment; mix on medium-high speed until pale and fluffy. Mix in eggs, 1 at a time, and vanilla. Reduce speed to low. Add flour mixture in 2 batches, alternating with the sour cream. Spread batter evenly over apples.
Bake until cake is golden brown and a cake tester inserted into center comes out clean, about 45 minutes. Let cool on a wire rack 15 minutes. Invert onto a serving plate.