Preheat the oven to 350°F (175°C). Lightly grease a 9-inch round cake pan or line it with parchment paper.
In a large mixing bowl, combine the peanut butter, brown sugar, granulated sugar, unsweetened applesauce, and vanilla extract. Mix well until smooth and creamy.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring until just combined. Do not overmix.
Fold in the vegan chocolate chips evenly into the dough.
Transfer the dough into the prepared cake pan and spread it out evenly with a spatula.
Bake in the preheated oven for 22 to 25 minutes, or until the edges are golden and a toothpick inserted in the center comes out with a few moist crumbs.
Remove from the oven and allow the cookie cake to cool completely in the pan on a wire rack before slicing and serving.