Preheat grill or grill pan. Lightly brush naan with olive oil on both sides. Grill for 2-4 minutes, flipping once part way through, or until grill marks are nicely defined and bread is heated through. (Tip: if using a grill pan, try placing a weight on top of the bread, such as a smaller sized skillet or a brick wrapped in foil, which will press the bread firmly into the grill pan and produce more defined grill marks).
Cut burrata into chunks, and spread onto warm flatbread. Top with sliced plums, baby arugula, and torn basil.
Drizzle with balsamic glaze and sprinkle liberally with sea salt and black pepper to taste.