Bring a large pot of salted water to a boil.

Heat olive oil in a large skillet over medium-low. Add garlic and cook until fragrant, about 2 minutes.

Increase heat to medium and add tomatoes. Cook, stirring occasionally, until most of the tomatoes break down and become jammy, about 10 to 12 minutes. Season with red pepper flakes, salt, and black pepper.

Reduce heat to medium-low and stir in heavy cream.

Add gnocchi to boiling water and cook according to package instructions until al dente. Reserve 1/2 cup pasta cooking water and drain remaining.

Transfer cooked gnocchi and reserved pasta water to skillet with tomato sauce and toss continuously until the sauce is thickened and glossy, about 2 minutes. Stir in fresh basil.

Break burrata balls into pieces and disperse evenly over pasta. Remove skillet from heat and garnish with reserved basil.