Preheat your oven to 350. Lightly oil a 9 x 13 baking dish or jelly roll pan (you can also use 12 x 8 dish/pan).
Whisk together the flour, baking powder, baking soda, and salt in a large mixing bowl.
In another bowl, whisk together the applesauce, peanut butter, brown sugar, vegan butter or oil, non-dairy milk, and vanilla. Add these wet ingredients to the dry and mix until they're evenly incorporated. The batter will be on the thick side. Pour it into the prepared baking dish and use a spatula or inverted spatula to spread it around evenly. Dot the jam in big spoonfuls on top, then use your spatula to swirl everything around. It doesn't have to be neat! Have fun.
Bake the cake for 35-40 minutes, or until the top of the cake is firm and the edges are lightly golden. Allow the cake to cool for at least 3 hours before slicing and serving. Enjoy!