Preheat oven to 350°F. Grease a 9-inch springform pan.
Make the Graham cracker base: mix together Graham cracker crumbs, 1/3 cup sugar, and melted butter until well-combined. Press mixture into bottom of prepared pan.
Make the filling: beat together cream cheese and 1/2 cup sugar until smooth. Beat in eggs, one at a time. Add rosemary and vanilla and mix until combined. Fold in chopped figs. Pour mixture into crust.
Bake for 45 to 50 minutes, or until edges are set and center is slightly jiggly. Let cool completely in pan, then refrigerate for at least 2 hours.
Make the caramel sauce: in a heavy-bottomed saucepan, melt 1 cup sugar over medium heat. Stir constantly until sugar melts and turns amber in color, about 5 minutes. Remove from heat and whisk in butter until melted. Add cream and salt and whisk until smooth.
Remove cheesecake from pan and place on serving platter. Drizzle caramel sauce over top. Serve chilled.