Combine figs with lemon zest and juice in a medium saucepan. Cover and cook over medium heat, stirring occasionally, until the figs are softened and shapeless, about 10 minutes.Stir in sugar and continue to cook, stirring frequently, until thickened and jam-like, about 10 minutes more. Remove from heat and let cool to lukewarm, then purée in a blender or food processor with the heavy cream and vanilla.Cover and refrigerate at least 2 hours or overnight, then churn according to manufacturer’s instructions.Meanwhile, gently melt chocolate in a small saucepan over low heat or in the microwave in 15 second bursts. When chocolate is melted and smooth, transfer to a zip-top bag and seal well, pressing out as much air as possible. If necessary, place bag in a bowl of warm water to keep warm while the ice cream finishes churning.When ice cream is the consistency of soft serve (about 1-2 minutes before being completely churned), cut 1/4 inch off the corner of the bag. Slowly drizzle most of the chocolate into churning ice cream, allowing the chocolate to swirl throughout. Transfer to a freezer safe container, drizzling a bit of remaining chocolate on top, and freeze at least 2 to 3 hours or overnight until firm.