Preparation

Make the vinaigrette:

• In a small bowl, whisk together the vinaigrette ingredients, seasoning to taste with the salt and pepper.

Make the salad:

• Position a rack 6 inches from the broiler element and heat the broiler on high. Arrange the bread slices on a baking sheet and broil until crispy and light golden on top, 1 to 2 minutes. Flip and broil the other sides until golden, about 1 minute. Rub one side of the toasted bread with the cut sides of the garlic. Sprinkle each slice with a small pinch of salt and set aside.
• Heat the oil in a 10-inch skillet over medium-high heat. When the oil is shimmering, add the mushrooms and stir to coat in the oil. Let the mushrooms cook undisturbed until the liquid they release evaporates and they’re deep golden brown, 5 to 7 minutes. Sprinkle with 1/2 tsp. salt, stir, and continue cooking, stirring occasionally, until most sides are nicely browned, 3 to 5 minutes more. Season to taste with more salt and pepper. Remove from the heat and let cool slightly.
• Toss the spinach and vinaigrette in a large bowl. Put the bread slices on four or six plates and drizzle each slice with a little olive oil. Divide the spinach among the plates, arranging it on the top of the bread but leaving part of the bread exposed. Top with the mushrooms. Using a cheese shaver or vegetable peeler, shave a few thin slivers of Parmigiano over the top. Serve immediately.