Preheat your oven to 400°F (200°C). In a large mixing bowl, combine the cherry tomatoes, chopped garlic, stripped thyme leaves, olive oil, kosher salt, and black pepper. Toss the mixture until the tomatoes are well-coated, then spread them evenly on a baking sheet. Roast them in the oven for approximately 30-40 minutes, or until they are tender and caramelized, stirring halfway through for even cooking.

While the tomatoes are roasting, gently open the burrata balls with a sharp knife, taking care not to pierce the outer skin too deeply. Using a spoon, fill each burrata generously with the vibrant pesto, letting it ooze slightly from the center for an enticing look. Set the stuffed burrata aside on a serving platter.

Once the cherry tomatoes are beautifully roasted, remove them from the oven and let them cool slightly. Arrange the pesto-stuffed burrata on the platter alongside the warm, caramelized tomatoes. Drizzle extra-virgin olive oil over the top, and finish with flaky sea salt and fresh basil leaves for a gorgeous presentation.