Working with one wrapper at a time, and keeping the others covered with a barely-damp cloth, wet the edges of the wrapper. Add approximately 1 T filling to the center of the wrapper, being careful not to overstuff.
In a large skillet, heat oil. Add mushrooms and garlic to skillet. Cook for 2-3 minutes. Add cabbage and carrots to skillet. Saute until cabbage and carrots have wilted and softened, 5-7 minutes. Add ginger and green onions, cook for 1 minute. Set aside and cool. Season with salt and pepper, toss with sesame oil and cilantro.