Separate the caps and stems of the king oyster mushrooms. Slice the caps with a knife and shred the stems with a fork. Chop the base of the shimeji mushrooms and ensure they are separated. In a large lidded container, combine 1/2 tbsp gochujang, 2 tsp tamari, 1/2 tbsp maple syrup, and 1 tsp rice vinegar, stirring until a homogenous paste is formed. Add the mushrooms to the container, close the lid, and shake well to coat them evenly. In a large pan over medium heat, add the mushrooms and cook for about 10 minutes until they release their juices. Once they start looking dry, add the sesame oil, stir, and cook for 2 minutes without stirring to allow browning. Stir again, then leave them undisturbed for another 2 minutes to brown further.