In a small bowl, whisk together the flour, oats, baking soda, salt, and cinnamon. This ensures the leavening agent and spices are evenly distributed throughout the batter, which is especially important in a single-serving mug cake where mixing is minimal. Set this dry mixture aside.
In another small bowl or mug, mash the banana until completely smooth—this creates a more uniform texture throughout the bread. Stir in the brown sugar, sugar, and vanilla extract until well combined. I like to make sure the banana is really broken down here because any lumps won't fully cook through in the short microwave time.
Add the dry ingredient mixture from Step 1 to the wet ingredients from Step 2 and stir until just combined—don't overmix, as this can lead to a dense, tough crumb. Transfer the batter to a microwave-safe mug (an 8-12 oz mug works well) and microwave on high power for 3-4 minutes. At the 3-minute mark, insert a toothpick into the center; if it comes out clean or with just a few moist crumbs, the bread is done. If needed, continue microwaving in 15-second intervals. I find that stopping at exactly 3 minutes sometimes leaves it slightly underdone, so don't skip the toothpick check.
Let the mug bread cool for 1-2 minutes before eating—this allows the crumb structure to set slightly and makes it easier to eat with a spoon. Enjoy warm, directly from the mug!