Add the cherry tomatoes and onions to a bowl. Add 1 tablespoon of olive oil, 1 tablespoon of balsamic vinegar, lemon juice, and ½ teaspoon of salt to the tomatoes. Toss until the tomatoes are coated. Add basil and toss again. Set aside.
Bring a large pot of salted water to a boil. Add pasta and cook until al dente, about 5-7 minutes.
Set aside about ½ cup of pasta water before straining the pasta.
Add the pasta back into the pot and add 1 tablespoon of olive oil, 2 tablespoons of balsamic vinegar, and 2 tablespoons of pasta water to the pasta. Toss to coat.
Heat the pasta over medium heat until the pasta becomes glossy and there is no liquid at the bottom of the pan. Add the remaining salt and pepper and toss again.
Transfer the pasta to a platter or large pasta bowl. Top with the tomato mixture. Break apart both burrata balls and place them on top of the tomatoes.
Garnish with fresh basil, fresh pepper, red pepper flakes, and a sprinkle of coarse sea salt (if desired).