Preheat an outdoor grill for high heat, and lightly oil the grate.
Place figs, skin-side down, on grill. Cook until figs start to swell and bubble, 2 to 4 minutes.
Whisk together balsamic vinegar and olive oil until completely combined, about 1 minute.
Place Burrata cheese on top of each piece of toast. Spread figs around bread and drizzle with balsamic vinegar mixture. Season with sea salt and black pepper to taste.
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