Peanut Butter Cake

Preheat oven to 350 degrees F (175 degrees C). Grease a baking sheet. Whisk flour, white sugar, and baking soda together in a bowl. Beat 1/2 cup milk, eggs, and 1 teaspoon vanilla extract into flour mixture using an electric mixer on low until smooth. Cook and stir water, 1/2 cup butter, vegetable oil, and 1/2 cup peanut butter together in a saucepan over medium heat until smooth, about 5 minutes. Stir peanut butter mixture into flour mixture until batter is well mixed; pour onto the prepared baking sheet. Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, about 20 minutes. Melt 1/2 cup butter and 1/2 cup peanut butter together in a saucepan over medium heat until smooth; add 1/3 cup milk and 1 teaspoon vanilla extract. Whisk confectioners' sugar into mixture until icing is smooth, 3 to 5 minutes. Pour icing over warm cake.