Closely adapted from Eric Silverstein.
Heat a large and wide skillet (or pot as long as it is wide) over medium-high heat. Add the oil when it is hot.
Add the onions with a pinch of salt. Stir and cook for 2-3 minutes.
Reduce heat to medium and add the miso and brown sugar. Stir to evenly incorporate. Let cook for 15 minutes, stirring occasionally, until the onions are golden brown.
Add 1/4 cup of the sake. Stir and reduce the heat to medium low and continue cooking for 20 minutes. The onions will start to really caramelize, so stay close and keep stirring to prevent scorching. The onions should become quite brown.
Deglaze the pan with the remaining sake and stir the onions for 5 more minutes. When the wine has cooked off, remove the pan from the heat. If the onions are in danger of scorching, scoop them out of the pan and into a serving dish. Set aside for the burgers.
The onions will keep up to 5 days in a sealed container in the fridge.