Make the balsamic glaze: Add the balsamic vinegar and honey to a small saucepan over medium-high heat. Bring to a gentle boil and cook for 3 minutes. Reduce the heat to low and simmer for about 10 minutes, stirring occasionally, until reduced by about half and thick enough to coat the back of a spoon. Remove from the heat and let cool; it will continue to thicken as it sits.
Make the pesto: Add the basil, Parmesan, pine nuts, garlic, olive oil, and salt to a food processor or blender. Blend until finely chopped, scraping down the sides as needed. Add a little more olive oil, if necessary, until the pesto is spoonable but still thick.
Season the tomatoes: Arrange the tomato slices on a serving platter. Sprinkle them lightly with some of the flaky sea salt and let them sit for 5 minutes. This draws out their juices and makes the tomatoes taste sweeter and more flavorful.
Add the burrata: Tear the burrata into large, creamy pieces and arrange them over the tomatoes.
Finish the salad: Spoon dollops of pesto over the tomatoes and burrata. Drizzle with the balsamic glaze and a little good olive oil. Finish with the remaining flaky salt, black pepper, and fresh basil leaves. Serve immediately.