Grease a 13x9-inch baking pan with cooking spray. Line the bottom of the cake pan with a piece of parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, or a mixing bowl with a hand mixer, cream the butter, granulated sugar, and brown sugar on medium speed until fluffy, 3 to 5 minutes. Add the oil and peanut butter and beat, scraping down the sides of the bowl as needed, until smooth and well combined. Scrape down the sides of the bowl again, then beat in the eggs, one at a time. Beat in the vanilla extract.

In a medium bowl, whisk together the flour, baking powder, salt, and baking soda. On low speed, mix one-third of the flour mixture into the peanut butter mixture. Then mix in half of the milk, another third of the flour mixture, and the remaining milk. Add the remaining flour mixture, mixing until just combined. Do not overmix.

Scrape the batter into the baking pan. Smooth the top with a spatula if needed. Bake until a toothpick inserted in the center comes out clean, 36 to 40 minutes. Remove from the oven and let cool on a wire rack until the baking pan is no longer warm, about 1 hour.

In a mixing bowl, beat together the butter, peanut butter, and half of the powdered sugar. Mix in the heavy cream or whole milk and vanilla extract. Mix in the remaining powdered sugar, beating until smooth and fluffy. Using a spatula, spread the frosting over the cooled cake. Store leftovers covered or in an airtight container at room temperature for up to three days. Love the recipe? Leave us stars and a comment below!