To make the fig purée: Preheat your oven to 350°F.
Place the figs in a 9"x9" metal baking pan and pour the balsamic vinegar over top.
Bake for 12 minutes, stirring twice to prevent burning.
Remove the figs from the oven and let cool slightly, about 10 minutes.
Pour the figs and remaining liquid into the blender and purée until fully blended. Store in an air-tight container in the refrigerator for up to 1 week.
To make the cocktail: Fill cocktail shaker with ice.
Add fig purée, bourbon, lemon juice, maple syrup, and orange bitters. Shake for 15 seconds, then strain into a cocktail glass.