Fill a large pot with water and season it with enough salt so it tastes like the ocean. I usually do 2 tablespoons. Bring it to a boil, add the pasta, and cook until al dente, following the package instructions. Reserve ½ cup of the cooking liquid, then drain the pasta and set it aside.
Meanwhile, heat olive oil in a large skillet over medium heat until shimmering, 3-4 minutes.
Add the garlic and sauté, stirring constantly, for 20-30 seconds, taking care not to burn it.
Carefully add the cherry tomatoes, salt, and pepper, and stir to ensure they are thoroughly coated with the olive oil.
Turn the heat down to medium-low and simmer for about 10-12 minutes, stirring occasionally, until the tomatoes soften and begin to burst when pressed with the back of a wooden spoon. The sauce should be slightly thickened.
Off the heat, stir in the balsamic vinegar, taste, and add more if needed.
Add the cooked pasta, Parmesan cheese, and basil to the tomatoes, then toss to combine, ensuring the tomatoes are evenly distributed. If needed, use the reserved pasta water to loosen the sauce.
Tear the burrata into large chunks, add it to the pasta, and gently toss to distribute everything evenly.
Garnish with more basil and a sprinkle more of Parmesan cheese, and serve while it is still warm.