Oil a 6-inch nonstick skillet and heat it over medium heat.
In a small bowl, whisk the eggs together. Pour half of the eggs into the skillet, and cook without stirring until the omelet starts to brown and sizzle and is mostly cooked through, about 3 minutes. Sprinkle half of the mozzarella cheese over the omelet, cover the skillet with a lid, and cook for about 1 minute so the cheese melts. Transfer the omelet to a plate. Repeat with the remaining eggs and cheese to make a second omelet.
In a medium bowl, toss the cherry tomatoes with the arugula and the lemon dressing. Season with salt to taste, and top the omelet with the salad. Garnish with the shaved Parmesan cheese and prosciutto, if using.