Preheat the oven to 375 degrees F (190 degrees C). Lightly grease a 1 1/2- to 2-quart baking dish. Set aside.

Combine flour, brown sugar, peanut butter, canola oil, baking powder, vanilla extract, and salt in a large mixing bowl. Beat using a hand mixer on medium speed to incorporate ingredients, about 30 seconds. Beat in milk slowly and mix for 1 1/2 minutes more. Spread batter evenly into the prepared dish using a rubber spatula. Place the dish onto a larger baking pan with sides at least 1/4-inch deep to catch potential spills.

Microwave water in a microwave-safe measuring cup until boiling, 2 to 3 minutes. Add sugar, cocoa, vanilla extract, and salt; stir until sugar and cocoa have dissolved. Pour onto top of cake batter; do not stir. Carefully place into the oven.

Bake in the preheated oven until cake is set and a toothpick inserted into the center comes out clean, about 40 minutes. Remove carefully from the oven and place onto a cooling rack to cool and thicken up, at least 20 minutes, before spooning out.