1. Make topping: In a bowl, stir together flour, brown sugar, salt and cinnamon. Add butter and combine with fingertips until mixture is crumbly. Refrigerate while making batter.
2. Make cake: Preheat oven to 350ºF. Butter and flour a 9-by-13-inch cake pan. In a small bowl, combine flour, baking powder, baking soda, salt and spices. In a large bowl, whisk together oil, applesauce, pumpkin, vanilla and both sugars until well mixed. Whisk in eggs. Add flour mixture to pumpkin mixture and stir until just combined; do not overmix. Pour batter into cake pan and spread evenly. Sprinkle with reserved topping.
3. Bake cake until a toothpick inserted into center comes out clean, 45 to 50 minutes. Allow to cool in pan on a wire rack at least 20 minutes.