Preheat the oven to 400 degrees. Place the tomatoes, onions, and garlic on a baking sheet and toss with the olive oil and salt. Roast in the oven for 25-30 minutes, stirring every 10 minutes, until tomatoes have burst and onions are browned on the edges.
Meanwhile, cook the pasta according to package directions, adding a few pinches of salt to the cooking water. If the pasta is finished before the tomatoes, drain, return to the pot, drizzle with olive oil, and cover to keep warm.
Toss the roasted vegetables and all the juices from the pan with the warmed pasta. Divide among four plates. Top each mound of pasta with half a ball of burrata. Allow the cream to melt over the pasta. Sprinkle with basil, salt, pepper, and a final drizzle of olive oil. Serve immediately.