Carefully peel figs and trim and discard stems. Stir together sugar, dark brown sugar, salt, and water in large heavy-bottomed saucepan or Dutch oven over medium-high heat and boil syrup until sugars are dissolved and syrup thickens, 5 to 7 minutes. Reduce heat to medium-low and add figs. Cut a piece of parchment paper in the shape of pot. Cut a nickel-sized vent in the middle. Press parchment directly onto surface of fig-syrup mixture. Simmer figs until translucent, gently stirring from time to time, about 1 hour. Cool figs completely, at least 1 hour. Figs may be served at room temperature or chilled.
