In a microwave-safe mug, add the unsalted butter. Microwave for about 20-30 seconds, or until melted.
To the melted butter in the mug, add the granulated sugar and unsweetened cocoa powder. Stir until well combined and smooth.
Crack the egg into the mug and add the vanilla extract and a pinch of salt. Stir until the mixture is smooth and all ingredients are fully incorporated.
Place the mug in the microwave and cook on high for about 60-90 seconds, or until the brownie is set but still slightly gooey in the center. Cooking time may vary depending on your microwave's wattage, so adjust accordingly.
Carefully remove the mug from the microwave (it will be hot!) and let the brownie cool for a minute or two.
If desired, sprinkle the top of the brownie with powdered sugar, add a dollop of whipped cream or vanilla ice cream, or sprinkle with chocolate chips.
Enjoy your delicious flourless chocolate mug brownie straight from the mug with a spoon!