Step 1

Combine vinegar, soy sauce (or Worcestershire) and sage in a small bowl. Place mushroom caps, gill-side up, in a large shallow dish and pour the vinegar mixture into them. Let marinate for about 10 minutes.

Step 2

Meanwhile, heat 1 tablespoon oil in a medium skillet over medium heat. Cook onions, stirring often, until just starting to brown, about 10 minutes. Add water and cook until very soft, 10 to 15 minutes more. Season with salt and pepper and set aside.

Step 3

Preheat oven to 250 degrees F.

Step 4

Place breadcrumbs on a large plate. Make sure the mushrooms are coated on both sides with marinade; dredge them in the breadcrumbs.

Step 5

Heat 1 tablespoon oil in a large nonstick skillet over medium heat. Add 4 mushrooms, gill-side down. Place a heavy, heatproof plate on top and cook until the breadcrumbs are crispy, pressing on the plate periodically to flatten the mushrooms, about 4 minutes. Remove the plate; add 1 tablespoon oil to the pan and turn the mushrooms over. Replace the plate and cook, pressing once or twice, until the breadcrumbs are golden brown and the mushrooms are tender, about 4 minutes more. Transfer to a baking sheet and keep warm in the oven. Wipe out the pan and repeat with the remaining 4 mushrooms and 2 tablespoons oil, reducing the heat if necessary. Serve with the onions and blue cheese.