Bring a large pot of salted water to a boil. Cook gnocchi according to package directions.
Heat olive oil in a large skillet over medium heat. Use a slotted spoon to transfer gnocchi to skillet, reserving ½ cup cooking liquid. Season with salt and pepper and cook until golden and crispy, about 3 minutes per side. Transfer to a large plate.
Melt butter in the same skillet over medium-low heat. Cook, stirring frequently, until butter begins to brown and smell nutty, about 5 minutes. Add garlic and cook until fragrant, about 30 seconds. Return gnocchi to skillet, along with about ¼ cup pasta cooking liquid and parmesan. Cook over medium-high heat, tossing and adding more pasta cooking liquid as needed, until a light sauce coats the gnocchi, about 2 minutes.
Season with salt and black pepper, stir in basil, and remove from heat. Tear burrata into serving bowls, top with gnocchi and garnish with more torn basil. Serve hot.