Preheat oven to 375 degrees F (190 degrees C). Grease 2 muffin tins with cooking spray.
Combine all-purpose flour, brown sugar, white sugar, whole wheat flour, baking soda, cinnamon, allspice, salt, and baking powder in a large bowl. Mix in pumpkin puree and apples.
Pour egg replacer into a bowl. Stir in applesauce gradually to avoid lumps. Whisk in oil. Add to the flour mixture; mix well until batter is smooth.
Spoon batter into the greased muffin cups, filling each one almost to the top.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 25 minutes. Cool in the pan for 5 minutes. Transfer to a wire rack to cool completely.
