Heat the broiler with a rack about 6 inches from the heat source.
STEP 2
In a large skillet, heat olive oil over medium-high. Add half the gnocchi, cover, and cook until golden brown, 2 to 4 minutes. Transfer to a bowl and repeat with remaining gnocchi.
STEP 3
Add butter to the skillet over medium-high, stirring until golden-brown, 1 to 2 minutes. Incorporate garlic, red-pepper flakes, salt, pepper, and 3 tablespoons water. Add tomatoes and cook until softened, 4 to 6 minutes, smashing them as they burst.
STEP 4
Stir in gnocchi and basil, then shake into an even layer. Top with mozzarella, drizzle with olive oil, and broil until melted, 2 to 4 minutes. Serve with more basil, red-pepper flakes, and black pepper.