Preheat your oven to 350°F and grease an 8-inch square baking pan to prevent sticking.

Combine the flour, brown sugar, cinnamon, baking powder, baking soda, and salt directly in the prepared baking pan. Use a fork to thoroughly blend the dry ingredients, ensuring an even distribution.

Form a well in the center of the dry mixture using your hands. Crack the egg into the well, then add the olive oil, yogurt, applesauce, and vanilla extract. Gently mix everything together with a fork until the batter is smooth and well-combined. Smooth the surface of the batter evenly with the edge of a butter knife for a consistent bake.

Place the pan in the oven and bake for 25 to 30 minutes. The cake is ready when the center feels firm to the touch and a toothpick inserted into the middle comes out clean, with no wet batter clinging to it.

Remove the cake from the oven and let it cool completely in the pan on a countertop, approximately 1 hour. Once cooled, dust the top generously with powdered sugar for a decorative and flavorful finish. Cut the cake into pieces and carefully lift them out using a small spatula.

Store the cake at room temperature for up to 3 days, or refrigerate it for up to 5 days to maintain its freshness.