Trim the base of each tomato so they sit flat. Using a serrated knife, slice off top ¼ inch of each tomato. Scoop out seeds and pulp, then place hollowed-out shells upside-down on a paper-towel-lined plate or baking sheet to drain, about 5 minutes.
Turn tomatoes upright, and season with salt. Open burrata and pour creamy interior into a bowl. Chop firm exterior into ½-inch pieces, then stuff one piece in the bottom of each tomato. Divide creamy center among tomatoes (about 1 teaspoon each). Drizzle tomatoes with oil, sprinkle with salt and pepper, and garnish with basil. Serve immediately.
