For sauce, stir miso, butter, garlic, mirin, honey, and rice vinegar together in a bowl. Set aside.
Heat a skillet over medium-low heat. Add oil and let it warm. Add mushrooms and salt. Stir and let mushrooms cook until they begin to soften, about 5 minutes.
Pour in miso sauce. Stir, then cook without stirring until caramelized, 10 to 15 minutes.
Once golden brown, remove to a plate. Season with salt and garnish with chives.
