Grease a 6 × 3-inch round baking pan with softened butter; set aside.
In a large mixing bowl, combine the melted butter, flour, 3/4 cup of the brown sugar, milk, 1/4 cup of the cocoa, eggs, baking powder, and vanilla. Using a hand mixer, beat until smooth. Pour the batter into the pan and sprinkle with the remaining 1/2 cup brown sugar and 1/4 cup cocoa powder.
Pour 1 1/2 cups water into the Instant Pot. Place a trivet in the pot. Place the pan on the trivet. Carefully pour the boiling water on top of the batter. Do not stir.
Secure the lid on the pot. Close the pressure-release valve. Select MANUAL/PRESSURE COOK and set the pot at HIGH pressure for 20 minutes. At the end of the cooking time, allow the pot to sit undisturbed for 10 minutes, then quick-release any remaining pressure.
Serve warm with ice cream or whipped cream, if desired.