Cook the tortellini according to package directions until just tender. Drain and rinse briefly under cold water, then drain thoroughly.
In a large bowl, whisk together mayonnaise, sour cream, red wine vinegar, pepperoncini brine, lemon juice, olive oil, grated Parmigiano-Reggiano, minced garlic, Dijon mustard, Italian seasoning, salt, black pepper, and red pepper flakes until smooth.
While the tortellini is still slightly warm, toss it with half of the dressing. Let it cool for about 10 minutes so the pasta absorbs the flavors.
Add the salami, pepperoni, mozzarella, provolone, tomatoes, celery, red onion, pepperoncini, giardiniera, oregano, parsley, and remaining grated Parmigiano-Reggiano. Toss gently until evenly combined.
Cover and refrigerate for at least 2 hours, preferably overnight, to allow the flavors to meld together.
Just before serving, gently fold in the shredded lettuce, fresh basil, and the remaining dressing. Transfer to a serving bowl and finish with shaved Parmigiano-Reggiano, basil ribbons, pepperoncini slices, and a drizzle of olive oil if desired.
