Mix balsamic vinegar with brown sugar in a saucepan over medium heat, stirring constantly until sugar has dissolved. Bring to a boil, reduce heat to low, and simmer until glaze is reduced by half, about 20 minutes. Glaze should coat the back of a spoon. Let cool.
Cut figs into quarters and slice the Ash Brie. Toast sliced bread, top with prosciutto, figs and cheese. Drizzle the balsamic glaze and sprinkle some thyme before serving.