In a medium bowl, stir together the flour, brown sugar, baking powder, salt, and cinnamon. In another small bowl, whisk together the eggs, butter, milk, and vanilla. Pour the wet ingredients into the bowl with the dry ingredients, and stir gently until just combined and the batter is just combined. Then fold in the grated apple with a spatula.
Heat about half a tablespoon of butter or olive oil in a non-stick frying pan or griddle. Add ¼ cup of the batter per pancake and cook for 2-3 minutes, or until bubbles start to form on the top of the pancake. Flip and cook an additional 1 minute, or until cooked through. Serve warm with butter and maple syrup.