Preheat oven to 400. Butter a baking sheet and slice plums in half. Arrange plums cut side up and sprinkle with brown sugar. Cook for about 15 min or until tender. Let cook for about 5 min and slice.
Heat sugar and water to a pan over high heat until the sugar dissolves. Bring to a simmer, and cook, swirling occasionally, until medium amber, about 4 minutes. Add hazelnuts and salt, and stir for 1 minute to coat.
Pour mixture onto a baking sheet lined with parchment and immediately separate nuts using 2 forks so that each stands alone on sheet. Let cool and roughly chop.
Coat bread in olive oil and cook in a pan for about 2-3 min per side. Add burrata. Top with sliced plums and chopped hazelnuts. Enjoy!