1. Preheat oven to 375 degrees, with rack in lower third of oven. Butter a 12-inch round tart pan with a removable bottom; line with a parchment round, and butter parchment. Wrap exterior of pan with foil.

2. With an electric mixer on medium-high speed, beat 1/2 cup (1 stick) butter, the brown sugar, Calvados, and a pinch of salt until pale and fluffy, 2 to 3 minutes. Spread mixture in prepared pan. Arrange apple rings in pan (you may not need all the apple rings; use the prettiest ones), and press them gently into mixture. Place cinnamon sticks in gaps. Sift together flour, baking soda, baking powder, remaining 3/4 teaspoon salt, and the ground cinnamon into a bowl.

3. With mixer on medium-high speed, beat remaining 3/4 cup (1 1/2 sticks) butter and the granulated sugar in a clean bowl until pale and fluffy, 2 to 3 minutes. Beat in eggs, 1 at a time, and vanilla. Reduce speed to low. Add flour mixture in 2 batches, alternating with the sour cream and beginning and ending with the flour; beat until just combined. Spread batter evenly over apples.

4. Bake until cake is golden brown and a cake tester comes out clean, about 45 minutes. Transfer to a wire rack to cool 15 minutes, then carefully invert onto a serving plate.