Heat some olive oil in a pan over medium heat. Add the mushrooms and white pepper. Allow the mushrooms to brown slightly before stirring. Let them cook down for 5 minutes. Transfer to a bowl and allow to cool for 5 minutes.
Add the rest of the dumpling ingredients to the bowl with the mushrooms and stir to combine.
To assemble, line a baking sheet with parchment paper. Take a won ton wrapper and brush the edges with water. Place 1 teaspoon of filling in the center and fold it in half to make a triangle. Be sure to press all of the air out and ensure the sides are fully sealed. Fold in one corner and then the other. Place the finished dumpling on the baking sheet. Repeat with the remaining ingredients.
Place the finished dumplings in the freezer for 10 minutes.**
While the dumplings are in the freezer, prepare the sauce. Place a dry pan on the stove over medium heat. Add the Sichuan peppercorns and toast in the pan until fragrant. Transfer to a spice grinder or mortar and pestle and grind into a fine powder.*** Add the ground peppercorns and the rest of the sauce ingredients to a bowl. Stir to combine and set aside.
To prepare the dumplings, bring a pot of water to a boil. Add the dumplings 5-8 at a time and cook for a couple of minutes, or until they float to the top - when you first add them to the pot, you can gently stir them to keep them from sticking to the bottom. Use a slotted spoon to remove the dumplings and transfer to a plate.
Drizzle the dumplings with the sauce and top with more green onion.