Preheat the oven to 350°F (175°C). Grease and flour a 9×13-inch sheet pan.
In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, and salt. Set aside.
In a large mixing bowl, beat the granulated sugar, brown sugar, peanut butter, and vegetable oil until smooth and creamy.
Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Fold in the shredded zucchini and 1/2 cup of the chopped peanuts.
Pour the batter evenly into the prepared sheet pan and spread it out smoothly.
Bake for 30 to 35 minutes, or until 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 frosting: In a medium bowl, beat the softened butter and peanut butter until creamy.
Gradually add the powdered sugar, alternating with milk, beating until smooth and spreadable.
Stir in the vanilla extract.
Spread the peanut butter frosting evenly over the cooled cake.
Sprinkle the remaining 1/2 cup chopped peanuts over the frosting for garnish.
Cut into 12 squares and serve.