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, 3/4 cup brown sugar, and 1/4 cup granulated sugar. Mix until smooth and creamy.
Add 1/4 cup unsweetened applesauce and 1 teaspoon vanilla extract to the peanut butter mixture. Stir until fully incorporated.
In a separate bowl, whisk together 1 1/4 cups all-purpose flour, 1/2 teaspoon baking soda, and 1/4 teaspoon salt.
Gradually add the dry ingredients to the wet ingredients, mixing until a soft dough forms.
Transfer the dough to the prepared cake pan and spread it evenly with a spatula.
Bake for 22-25 minutes, or until the edges are golden and a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool completely in the pan on a wire rack.
While the cake cools, prepare the peanut butter glaze by whisking together 1/2 cup powdered sugar, 2 tablespoons creamy peanut butter, and 1 tablespoon unsweetened almond milk until smooth.
Once the cake is completely cooled, spread the peanut butter glaze evenly over the top.
Slice and serve the vegan peanut butter cookie cake.