Heat the oven. Preheat to 425°F (220°C). Line a sheet pan with parchment for easy cleanup.
Prep the tomatoes. Toss cherry tomatoes with 3 tablespoons olive oil, salt, pepper, and red pepper flakes. Spread on the pan in a single layer.
Add the garlic slices, tucking some under the tomatoes so they don’t burn.
Roast until jammy. Bake 18–25 minutes, stirring once. The tomatoes should burst and caramelize around the edges. If they seem too tart, stir in sugar or honey, and the balsamic if using, during the last 3 minutes.
Boil the pasta. While tomatoes roast, cook pasta in well-salted water until just shy of al dente.
Reserve 3/4 cup of the starchy water before draining.
Make the sauce base. Transfer the roasted tomatoes and all their juices to a large skillet or back into the empty pot. Add 1/4 cup reserved pasta water to loosen and simmer on low for 1–2 minutes.
Toss the pasta. Add drained pasta to the tomatoes. Drizzle in the remaining 1 tablespoon olive oil and sprinkle in the Parmesan.
Toss and add more pasta water as needed to create a glossy, lightly saucy coating.
Finish with herbs. Fold in torn basil and a pinch of black pepper. Taste and adjust salt and acidity. A little lemon zest perks it up.
Plate with burrata. Twirl the pasta into bowls.
Tear the burrata into large pieces and nestle on top, or place the whole ball in the center. Sprinkle with flaky salt, extra Parmesan, and a few basil leaves.
Serve immediately. The burrata should ooze into the warm pasta. Eat right away for the best texture.