Heat the oil in a large skillet over medium-high heat. Add the mushrooms, green onions, garlic, and ginger and cook until the mushrooms have released their juices and the liquid has evaporated, 8-10 minutes. Stir in the tamari, mirin, and white pepper and cook until the sauce has thickened, about 2 minutes more. Remove from heat.
Preheat oven to 400ºF. Use an oil mister to spray a large baking sheet with oil (or line a baking sheet with parchment paper if you don't have a mister).
Fill a small bowl with water. On a dry work surface, lay one wonton wrapper with a corner pointing towards you. Drop ¾ tablespoon of mushroom filling on the bottom half of the wrapper. Dip your finger in the water and brush the edges of the bottom half of the wrapper to dampen slightly. Fold the top half down onto the bottom half to form a triangle and press to seal. Place the finished wonton on the baking sheet and repeat until you've used all of the filling (you should have enough for 25-30).
Spray or brush the tops of the wontons with additional oil and bake for 8-10 minutes, until lightly browned and crisp. While the wontons are baking, whisk together all of the dipping sauce ingredients in a small bowl. Serve the wontons with the sauce.
Notes
The sesame oil used for the wontons is plain sesame oil -- not toasted! If you don't happen to have it, any cooking oil with a high smoke point can be substituted.
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