In a food processor, add the chopped mushrooms, ground chicken or pork, sliced scallions, soy sauce or tamari, Mirin, sesame oil, ginger, and garlic; pulse until just combined, 4 to 6 pulses.
In a large skillet (nonstick), add the avocado oil or a neutral cooking oil and warm over medium heat.
Loosely shape into 12 (about 2-tablespoon) balls with wet hands and place in the skillet, spaced about 1 inch apart.
Fill a shallow bowl about halfway with hot tap water. Working with one piece of rice paper at a time, soak in hot water until pliable, about 20 seconds; place over 1 meatball and cinch the wrapper around the edges.
Increase the skillet to medium-high heat and once the oil begins to sizzle, cook for 1 minute. Cover and cook, undisturbed, for 3 minutes.
Uncover and cook until the bottoms and edges are golden brown.
Insert a thermometer into the thickest portion of the meatball; cook until it reaches 165°F, 5 to 6 minutes.
Remove from heat, add more sliced scallions, drizzle with dumpling sauce, and serve.