Slice the mozzarella into eight slices and then again in half crosswise, so that you have 16 pieces of cheese. Whisk the egg in a medium bowl with a pinch of salt and pepper. Put the breadcrumbs in another medium bowl. Working with a few pieces at a time, dip the cheese in the egg, turning to coat all sides of the cheese. Dredge the cheese in the breadcrumbs, pressing to help the crumbs adhere and cover the cheese as much as possible. Transfer the breaded cheese slices to a plate and refrigerate until ready to cook. You can prepare the cheese up to 1 hour ahead. Discard any leftover egg and crumbs.
In a small bowl, combine the tomatoes, basil, and garlic and season with 1/2 tsp. salt and a few grinds of pepper. Let the tomato mixture sit for 5 min. and then add 1/4 cup of the oil and the vinegar
Heat 1-1/2 Tbs. of the oil in a 10-inch nonstick skillet over medium-high heat. Put half of the cheese in the pan and cook until the breadcrumbs turn golden, 30 to 60 seconds. Use two forks to turn the cheese and cook until the second side is golden, another 30 to 60 seconds. Transfer the cheese to a plate. Using the remaining 1-1/2 Tbs. oil, repeat with the second batch of cheese.
Put the arugula in a large bowl. Stir the tomato mixture and toss it with the arugula. Taste and add salt and pepper as needed. Portion the salad among four plates. Arrange four pieces of cheese on top of each salad and serve immediately.