Whisk soy sauce, brown sugar, garlic, ginger, rice vinegar, and sesame oil together in a nonreactive container and set aside until ready to use.

Heat 1 tablespoon of the canola oil in a large frying pan over medium-high heat. When the oil shimmers and wisps of smoke come off the surface, add the onion. Cook, stirring rarely, until translucent and charred, about 3 minutes. Remove to a large bowl and return the pan to the stove.

Add another 1 tablespoon of oil and add 1/3 off the mushrooms. Cook, stirring occasionally until browned and all liquid is cooked off, about 8 minutes. Add mushrooms to onion mixture and repeat twice more to cook off all remaining mushrooms.

Meanwhile, place half the marinade in a small saucepan and cook over medium high heat until reduced by half, about 3 minutes.

Add the remaining (non-reduced) bulgogi marinade to the last batch of mushrooms, add reserved mushrooms and onions and stir to coat through.

Cook until marinated is reduced and the mushrooms are coated a light glaze, about 5 minutes. Remove from heat, add reduced marinade and scallions and stir through.

To serve, place a few spoonfuls of brown rice in a lettuce leaf, add a few spoonfuls of mushrooms, then top with any of all of the desired toppings.