Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs for al dente. Reserve 3/4 cup cooking water and then drain.
Meanwhile, heat 3 tablespoons olive oil in a large pot or Dutch oven over medium-high heat. Add the garlic and cook until just starting to brown, about 1 minute. Add the tomatoes and a big pinch of salt and cook, without stirring, until the tomatoes start to blister and split, 1 to 2 minutes. Continue cooking, stirring and gently smashing the tomatoes, until they break down, 2 to 4 minutes. Stir in the red pepper flakes.
Add the pasta and 1/2 cup of the cooking water to the tomatoes. Cook over medium-high heat, gently stirring, until the pasta is well coated, 1 to 2 minutes. Add more cooking water if the pasta is dry; season with salt.
Divide the pasta among shallow bowls. Carefully cut the burrata into quarters and place on top of the pasta. Lightly season the burrata with salt and pepper and drizzle with the remaining 1 tablespoon olive oil. Sprinkle with the oregano.