Warm salads used to be common on restaurant menus, but their popularity has waned, for no good reason. This terrific salad is made with warm mushrooms that develop great meaty flavor when cooked with soy sauce.
In a large bowl, toss the mushrooms with the lemon juice. Preheat the oven to 300 degrees. In a large skillet, melt the buter in 2 tablespoons of the pure olive oil over moderately high heat. When the butter starts to brown, add the mushrooms and season with salt and pepper. Cover and cook until the mushrooms release their liquid, about 3 minutes. Uncover and continue to cook, stirring occasionally, until the liquid has evaporated and the mushrooms are deeply browned, about 8 minutes. Add the soy sauce and cook, stirring, for 2 minutes longer. Transfer the mushrooms to a rimmed baking sheet and keep warm in the oven.
In a small bowl, whisk the extra-virgin olive oil with the sherry vinegar, Marsala, tomato paste and minced garlic. Season the sherry vinaigrette with salt and pepper.
Add the remaining 1 teaspoon of pure olive oil to the skillet. Add the shallots, cover and cook over moderate heat until softened, about 3 minutes. Uncover and cook, stirring, until lightly browned. Stir in the sherry vinaigrette and remove from the heat.
In a bowl, toss the lettuces. Add the mushrooms and shallots and toss well. Arrange the salad on 6 plates, sprinkle with the cheese and serve at once.
Make Ahead: The vinaigrette can be made up to 3 hours ahead.