Combine all ingredients in a small saucepan and bring to a boil over high heat. Remove from heat, let cool to room temperature, then refrigerate for at least 30 minutes.
Heat grill to high. Brush mushrooms with oil and season with salt and pepper, to taste. Grill for 3 to 4 minutes per side or until just cooked through. Remove from heat and tear into rough pieces.
Place mizuna in a large bowl and toss with a few tablespoons of dressing. Season with pepper, to taste. Add mushrooms to bowl, toss, and drizzle with more of the dressing.