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Combine the onion, rice vinegar, and mint in a small bowl; toss and set aside to marinate.
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Preheat the oven to 450 degrees F.
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To make the dressing: Whisk together the cider vinegar, balsamic vinegar and mustard in a small bowl or glass measuring cup with a handle, and season with salt and pepper. Whisk in the oil in thin stream. Taste and adjust the seasoning if needed.
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To make the salad: Put the mushrooms on a rimmed baking sheet. Using a pastry brush, brush the mushrooms on both sides with dressing. Turn them stem-side up and roast until tender, about 15 minutes, brushing with dressing at 5-minute intervals, reserving the leftover dressing. Set them aside to cool slightly.
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Drain the onions, reserving the vinegar. Put the onions in a large bowl. Add the lettuce, bell pepper, and reserved dressing; toss. Taste and, if needed, season with the reserved rice vinegar, salt, and/or pepper. Divide the salad between six plates. Cut the mushrooms into 1/2-inch slices. Fan them on top of the salads. Top with the cheese.