Make the dressing: Whisk together all of the ingredients in a medium bowl and set aside.
Make the salad mix: Spin dry all the lettuces, celery, parsley, and onion in a salad spinner. Wrap in paper towels to dry thoroughly and refrigerate.
Meanwhile, halve the cherry tomatoes and olives and transfer to a very large bowl. Shave the cheese with a vegetable peeler to yield 1 cup. Reserve some of the shaved cheese for garnish and add the remaining cheese to the bowl.
Remove the vegetables from the paper towels and add to the bowl. Season with fleur de sel. Add the dressing and toss to coat. Taste and adjust the seasoning, if necessary. Scatter the salami and reserved cheese over the top and serve immediately. Alternatively, prepare the salad mix and refrigerate for up to 3 days. Toss each serving with some of the dressing, as needed.