Heat 2 tablespoons olive oil in a skillet over medium. Toss in the mushrooms along with a pinch of alt and pepper. Cook undisturbed for 5 minutes. Stir in the butter, garlic, and ginger for 2-3 minutes longer until fragrant and the mushrooms deepen to a golden hue. Move to a plate.
In a jar, shake together the soy sauce, brown sugar, rice vinegar, sesame oil, and ½ cup water. Pour into the skillet and boil for 3-4 minutes until slightly thickened.
In a small bowl, whisk together the corn starch and 1 tablespoon of water to make a slurry. Add the mushrooms and the slurry into the pan and stir until the sauce thickens and coats the mushrooms. Divide mushrooms into bowls with rice and garnish with green onions and sesame seeds, as desired.
