In a microwave-safe mug, combine the flour, granulated sugar, baking powder, salt, and 1/2 teaspoon of cinnamon. Mix well.
Add the milk, vegetable oil, and vanilla extract to the dry ingredients. Stir until smooth and well combined.
In a small bowl, mix the brown sugar and 1 teaspoon of cinnamon. Sprinkle this mixture over the batter in the mug. Use a knife or toothpick to swirl the filling into the batter gently.
Microwave the mug on high for 1 minute and 30 seconds, or until the cake has risen and is set in the middle. Cooking times may vary depending on your microwave, so check for doneness.
While the cake is cooling slightly, prepare the icing by mixing the melted butter, powdered sugar, and 1 tablespoon of milk until smooth. Drizzle over the warm mug cake.