Special equipment: Decorative toothpicks
Position an oven rack in the top third of the oven and preheat the oven broiler.
Remove the stems from the mushrooms and discard. Lay the mushroom caps gill-side-up and use a sharp knife to completely remove the gills (so that the caps will lie flat). Arrange the mushroom caps on a baking sheet, brush all over with 1 tablespoon of oil and sprinkle with 1/4 teaspoon salt. Broil until the caps are just tender, flipping halfway through, 4 to 5 minutes per side. Allow to cool completely.
Whisk together the vinegar, pepperoncini, oregano, remaining 1 tablespoon oil and a few grinds of black pepper in a small bowl.
Assemble the sandwiches: Arrange one mushroom cap, cut side-up, on a work surface. Fold 1 piece of provolone to fit on top of the cap and repeat with 1 slice each of ham and salami. Top with 1 slice of tomato and about 2 tablespoons of lettuce. Drizzle with some of the pepperoncini vinaigrette. Sandwich with another mushroom cap and secure with a toothpick threaded with an olive. Repeat with the remaining ingredients to make 3 more sandwiches. Wrap each sandwich halfway in wax paper (this will help catch all the juices) and serve.
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