Heat the broiler with a rack about 6 inches from the heat source.
In a large (12-inch) skillet on the stovetop, heat enough olive oil to lightly coat the bottom of the pan (about 1 tablespoon) over medium-high. Add half the gnocchi (1 package) to the pan, breaking up any that are stuck together. Cover with a lid or baking sheet and cook, undisturbed, until golden brown on one side, 2 to 4 minutes. Transfer to a medium bowl. Repeat with the remaining gnocchi and olive oil.
Add the butter to the skillet and cook over medium-high, stirring often, until golden-brown and toasty, 1 to 2 minutes. Add the garlic, red-pepper flakes, 1½ teaspoons salt and a few grinds of pepper, reducing the heat slightly if necessary to avoid scorching. Add the tomatoes and 3 tablespoons water and cook, shaking the pan occasionally, until the tomatoes have softened, and the liquid has slightly thickened, 4 to 6 minutes. Smash the tomatoes as they burst to help them along.
Add the seared gnocchi and ¼ cup basil (if using), stir to coat, then shake into an even layer. Top with the mozzarella and drizzle lightly with olive oil. Broil until the cheese is melted and browned in spots, 2 to 4 minutes. Top with more basil, red-pepper flakes and black pepper as desired.
