Combine all the ingredients for the vinaigrette in a small bowl and whisk to combine. Season with salt and pepper as needed.
Cut the iceberg lettuce in half through the core. Remove and discard the outer leaves and discard the core. Separate the lettuce leaves, stack 2 or 3 leaves on top of one another, then cut them lengthwise into 1/4 -inch-wide strips. Repeat with remaining leaves. thinly slice the radicchio in the same way.
In a large, wide bowl, combine the lettuce and radicchio, tomatoes, chickpeas, mozzarella, provolone, salami, and pepperoncini. Season with salt to taste and toss to thoroughly combine. Drizzle 6 tablespoons of the vinaigrette over the salad and toss gently to coat the salad evenly. Taste and add more vinaigrette as needed, plus salt and pepper. Transfer the salad to a large platter, sprinkle with extra oregano, and serve.