Generously grease an 8-inch round cake pan (no need to grease if pan is nonstick).
Melt the butter in a medium skillet over medium-high heat. Sauté the apples, tossing occasionally, until golden brown, 5 to 7 minutes. Stir in the Calvados, 1 tablespoon of the sugar, and the cinnamon. Set aside until needed.
Whisk together the yogurt and the remaining sugar in a large bowl, until very smooth, about 2 minutes. Combine the flour and baking soda, then add the oil and eggs, whisking well after each addition. Stir the apple pieces into the batter.
Pour the batter into the prepared pan, spreading it evenly. Set on a sturdy baking sheet and bake for 45 to 55 minutes, or until a toothpick inserted in the center comes out clean. Allow to cool on a wire rack.
Note: The cake is best made several hours before serving, or better yet, the night before.