Pulse mushrooms in a food processor until finely chopped, 10 to 12 pulses.
Add chicken, scallions, water, soy sauce, wine, sesame oil, ginger, garlic, sugar, white pepper; pulse until just combined, 4 to 6 pulses.
Loosely shape into 12 (about 2-tablespoon) balls with wet hands and place in a 12-inch nonstick skillet, spaced about 1-inch apart.
Fill a shallow dish about halfway with hot tap water, working with 1 rice paper square at a time, soak in hot water, until pliable, about 20 seconds; place over 1 meatball and cinch wrapper around the edges.
Drizzle with canola oil.
Place skillet over medium-high and once oil begins to sizzle; cook for 1 minute.
Cover, and cook, undisturbed, for 3 minutes.
Uncover and cook until bottoms and edges are golden brown and a thermometer inserted in the thickest portion of meatball registers 165°F, 5 to 6 minutes.
Remove from heat and drizzle with chile oil or chile crisp and sprinkle with additional scallions, to garnish.
Serve with soy sauce and chile crisp or oil stirred together in a small bowl.
