Preheat your oven to 350°F (175°C). Grease and flour a 9x13-inch baking pan.
In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
In a large mixing bowl, beat the softened butter and peanut butter together until smooth and creamy.
Gradually add the granulated and brown sugars, beating until fluffy.
Beat in the eggs, one at a time, followed by the vanilla extract.
Alternately add the dry ingredients and milk to the wet ingredients, beginning and ending with the dry mixture. Mix until just combined.
Pour the batter into the prepared pan, spreading it evenly. Bake for 30-35 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool completely before frosting.
To prepare the frosting: In a medium bowl, beat the peanut butter and softened butter together until smooth.
Gradually add the powdered sugar, mixing until combined. Stir in the vanilla extract.
Add milk, one tablespoon at a time, until the frosting reaches your desired consistency.
Spread the frosting evenly over the cooled cake.