Preheat the oven to 352°F (177°C) and position the rack in the middle of the oven.
Line the bottom of a 10-inch (25 cm) springform pan with parchment paper and lightly grease the sides with butter or oil.
In a small saucepan over medium heat, melt the butter. Take off the heat, and immediately whisk in the peanut butter and vegetable oil until evenly integrated. Leave in the pot to cool.
In the bowl of a stand mixer fitted with the whisk attachment, combine both sugars, baking powder, baking soda, salt, and eggs. Mix on low until just combined, then increase to high speed and beat until thick, fluffy, and pale—about 8 minutes.
Reduce the speed to low and slowly add the cooled peanut butter mixture. Mix until evenly incorporated.
Add the flour and mix until just combined. Scrape down the sides of the bowl, then add the buttermilk while the mixer is running on low.
Turn off the mixer and gently fold the batter a few times by hand to ensure it's smooth and fully combined.
Spread the batter out evenly in springform pan. Bake for around 30-35 minutes or until a cake tester comes out clean. Rotate halfway though the cooking time.
Set the cake aside to cool completely for at least 1 hour. Meanwhile, prepare the frosting.