Heat the oil in a large skillet. Add the mushrooms and saute until browned. Add the cabbage and saute until just slightly softened and fragrant (just a minute or two). Add the ginger, garlic, honey, mirin, vinger, sesame oil, radish, and green onion. Saute until most of the liquid is evaporated. Remove from the heat.
Using your finger dipped in water, gently wet around the four sides of the spring roll wrapper. Place 2 heaping tablespoons on a spring roll wrapper. Fold up one corner of the wrapper to the center, then the two side corners, then roll up to form the spring roll. (The package for my wrappers illustrated how to do this. Also, if you Google "how to fold a spring roll" and look at images, helpful photos pop right up.)
Heat a large rimmed pan with ~1/2-inch of canola oil. Working in batches, add spring rolls and fry, turning once, until nicely golden. Transfer to a plate with lined with paper towels.
In a small bowl, combine the ingredients for the dipping sauce. Serve with the spring rolls! Yields: ~21 servings (how many wrappers were in the package I had).