As soon as the crust finish baking, whisk the brown sugar, corn syrup, melted butter, bourbon, vanilla, and salt together until just combined. Whisk in the egg until incorporated. Stir in figs, then pour into the hot crust. Sprinkle the pecans over the top and gently press onto filling. Bake until the top is brown and cracks start to form across the surface, 25 to 30 minutes.