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 creamy peanut butter, brown sugar, and granulated sugar. Mix well until smooth and creamy.
Add the unsweetened applesauce and vanilla extract to the peanut butter mixture. Stir until fully incorporated.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until a thick dough forms.
Fold in the vegan chocolate chips evenly throughout the dough.
Transfer the dough to the prepared cake pan and spread it out evenly, smoothing the top 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 the cookie cake from the oven and allow it to cool in the pan for 15 minutes.
Carefully remove the cookie cake from the pan and transfer it to a wire rack to cool completely before slicing and serving.