Center the oven rack, and preheat the oven to 350°F (180°C). Grease a 9x5-inch loaf pan, and, if desired, line the bottom of the pan with baking parchment.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, and cinnamon.
In a large bowl, beat the egg with a whisk. Then whisk in the applesauce, oil, yogurt, brown sugar, and vanilla.
Add the dry mixture to the wet, and stir just until the flour disappears into the batter. Pour the batter into the prepared loaf pan and smooth the top.
Bake until the loaf puffs and browns, and when a skewer inserted into the center of the loaf comes out clean -- 45-55 minutes. Cool the bread in its pan for 10 minutes, and then unmold and cool completely before slicing.