Preheat the oven to 375 degrees F. Place the flatbread crust on a parchment-lined baking sheet.
In a small bowl, combine the hot pepper jelly with the balsamic vinegar. Spread the jelly mixture over the flatbread, leaving a border of crust exposed. Bake until the jelly begins to bubble, 4 to 5 minutes.
Top the flatbread with the figs. Tear the prosciutto into ribbons and place around the figs. Sprinkle with the baby arugula. Drizzle the olive oil over the flatbread and sprinkle with the flaky sea salt. Slice into triangular pieces and serve immediately.