Step 1: Preheat oven to 350°F (175°C). Line a 9x13 inch baking pan with parchment paper, leaving an overhang on the sides. Lightly grease the parchment paper.
Step 2: In a large mixing bowl, cream together the softened butter, granulated sugar, and light brown sugar with an electric mixer until light and fluffy (2-3 minutes).
Step 3: Beat in the egg and 1/2 teaspoon of the vanilla extract until well combined.
Step 4: Add 1 cup of the creamy peanut butter to the mixture and beat until smooth and fully incorporated.
Step 5: In a separate medium bowl, whisk together the all-purpose flour, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Step 6: Press about two-thirds of the peanut butter cookie dough evenly into the bottom of the prepared pan.
Step 7: In a medium bowl, whisk together the sweetened condensed milk, the remaining 1/2 cup of creamy peanut butter, and the remaining 1/2 teaspoon of vanilla extract until smooth.
Step 8: Pour the sweetened condensed milk mixture evenly over the cookie dough base in the pan, spreading gently with a spatula.
Step 9: Crumble the remaining one-third of the peanut butter cookie dough over the sweetened condensed milk layer.
Step 10: Bake for 25-30 minutes, or until the edges are lightly golden brown and the topping is set. The center may still appear soft.
Step 11: Remove from oven and let cool completely on a wire rack for at least 2-3 hours before lifting out of the pan and cutting into squares. Chilling for an hour can aid in cleaner cuts.