Preheat a grill pan or large nonstick skillet over medium-high or prepare an outdoor grill.
Place the meat in a bowl and season with salt and pepper. Add the egg yolk, cheese, tomato paste, parsley, garlic, and hot red pepper flakes. Mix the meat and form 12 large meatballs, 2 to 2 1/2 inches each. Flatten the meatballs and form patties. Drizzle the patties with EVOO and add them to the hot pan or grill. Cook for 3 minutes on each side. Hold the cooked patties on a plate under foil.
On a large platter or in a salad bowl, combine the romaine with the quartered artichoke hearts, roasted red pepper, hot peppers, olives, celery, salami, and provolone. Toss the salad with vinegar first, about 3 tablespoons, then toss with EVOO-up to 1/4 cup to your taste. (Just eyeball the amounts.) Season the salad with salt and pepper to taste. Scatter the chopped mozzarella, tomatoes, and basil across the salad. Arrange the cooked mini meatball burgers on top of the arranged antipasto salad.