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 1 cup creamy peanut butter, organic cane sugar, and brown sugar. Mix well until smooth and creamy.
Add the unsweetened applesauce, non-dairy milk, 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 into the batter.
Transfer the batter 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 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.
While the cookie cake cools, prepare the peanut butter glaze by whisking together powdered sugar, 2 tablespoons creamy peanut butter, and 1 tablespoon non-dairy milk until smooth and spreadable.
Once the cookie cake is completely cooled, spread the peanut butter glaze evenly over the top.
Slice into wedges and serve. Store any leftovers in an airtight container at room temperature for up to 3 days.