Bring a large pot of salted water to a boil over high. Stir in pasta; cook, stirring occasionally, until al dente, 12 to 14 minutes. Reserve about ¼ cup cooking water to thin pesto as needed. Drain pasta, and rinse with cold water. Let pasta drain completely, about 30 seconds.
While pasta is cooking, bring balsamic vinegar to a boil in a small skillet over low; cook, stirring constantly, until vinegar is reduced to about 1 tablespoon and thickens slightly, about 3 minutes. Set aside balsamic reduction.
Place cooked pasta in a large bowl; stir in pesto, sun-dried tomatoes and oil, cherry tomatoes, and 2 tablespoons reserved cooking liquid until well combined, stirring in remaining 2 tablespoons cooking liquid, as needed, to reach desired consistency. Add mozzarella and basil; toss gently to combine. Top with burrata cheese and basil leaves; sprinkle with flaky sea salt. Drizzle with reserved balsamic reduction. Serve.
