Bring a large pot of salted water to a boil and cook the pasta according to package instructions until al dente. Reserve about ¼ cup of pasta water, then drain.
While the pasta cooks, heat the olive oil in a large heavy skillet over medium heat. Add the cherry tomatoes and cook, stirring occasionally, until they begin to burst and soften, about 5 minutes.
Add the anchovy fillets and cook, pressing them into the oil until they dissolve completely.
Stir in the garlic and red pepper flakes and cook for about 30 seconds, just until fragrant. Gently crush some of the tomatoes with the back of a spoon, leaving others intact.
Add the diced tomatoes, balsamic vinegar, and sugar. Bring to a boil, then reduce to a simmer. Cook for 10-15 minutes, until the sauce thickens and flavors meld. Taste and adjust seasoning.
Add the drained pasta to the sauce along with the reserved pasta water. Toss over medium heat until the sauce is glossy and evenly coats the noodles.
Gradually add the reserved pasta water to the sauce, one splash at a time, stirring vigorously after each addition. Continue for 1-2 minutes, letting the starches thicken the sauce. Stop adding water once the sauce reaches a glossy, clingy consistency that will coat the pasta evenly.
Add the drained pasta to the sauce and toss until every strand is coated.
Divide among bowls. Tear the burrata over the top and finish with fresh basil and a drizzle of olive oil.