Heat the oven to 350°F. Grease a 9-inch square pan and line the bottom with parchment, leaving two handles.
Whisk flour, baking powder, baking soda, cinnamon and salt until no leavening lumps remain.
In a second bowl, whisk eggs and brown sugar for 45 seconds, then whisk in yogurt, oil and vanilla.
Fold dry ingredients into wet until a few flour streaks remain. Add apples and fold only until the batter is even.
Spread in the pan, level the top and add optional coarse sugar. Bake 40 to 50 minutes, rotating once after 30 minutes.
A tester inserted into a cake area should show moist crumbs, not wet batter. Cool 20 minutes in the pan, then lift out and cool at least 10 minutes more.