Bring a large pot of salted water to boil and cook gnocchi according to package directions, usually 2-3 minutes until they float.
Drain gnocchi well and heat 2 tablespoons olive oil in a large skillet over medium-high heat.
Add gnocchi to the skillet in a single layer and cook without stirring for 3-4 minutes until golden and crispy on the bottom, then flip and crisp the other side.
Add pesto and cherry tomatoes to the pan, toss everything together, and cook for 2 minutes until tomatoes soften slightly.
Transfer to a serving platter, tear burrata over the top, drizzle with remaining olive oil, and garnish with fresh basil and red pepper flakes before serving.