
Heat a large (about 12-inch), well-seasoned cast-iron or nonstick skillet over medium-high; add 2 tablespoons of olive oil. Add the gnocchi to the pan, breaking up any that are stuck together. Cook for 8 to 10 minutes, tossing every 1 ½ to 2 minutes so they get golden and crispy all over.
Meanwhile, prepare your tomatoes: If you are using small ones like cherry or grape varieties, simply slice them in half. For larger tomatoes, quarter them or slice into bite-size chunks. (It is good to have a mix of shapes and sizes.) Place the tomatoes and onions in a large serving bowl. Add the balsamic vinegar and 1 teaspoon salt; season with pepper and gently toss.
When the gnocchi are golden and crispy, add them to the tomatoes, along with 3 to 4 tablespoons of olive oil and a handful each of parsley and basil. Toss and taste, adding more salt or pepper as needed.
Top with more parsley, mozzarella and basil, and eat immediately or at room temperature.