Heat 2 tablespoons of the oil in a large skillet over high heat. Add the onion and cook until softened, about 5 minutes. Add the  prosciutto, tomatoes, garlic, basil, oregano, 1/2 teaspoon of the salt, and 4 turns of the pepper, and sauté for 2 minutes. Stir in the vinegar, remove from the heat, and turn into a large bowl.

Heat the remaining 2 tablespoons of oil in the same skillet over high heat.

When the oil is hot, add the tortellini and the remaining salt and pepper, and fry until golden brown, shaking the skillet and tossing the pasta, about 4 minutes. Remove from the heat and toss with the prosciutto in the bowl.

Add the greens and the Parmesan to the bowl and toss well.

To serve, divide the salad among 4 small plates or shallow pasta bowls. Sprinkle with additional Parmesan.