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 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 just combined. Do not overmix.
Fold in the vegan chocolate chips evenly throughout the dough.
Transfer the cookie 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 into the center comes out mostly clean with a few moist crumbs.
Remove the cookie cake from the oven and allow it to cool completely in the pan on a wire rack before slicing and serving.