Soak figs in a bowl of water until softened, 3 to 4 hours. Drain figs and chop.
Combine chopped figs, 2 tablespoons sugar, and lemon juice in a saucepan over medium heat; cook and stir until until sugar is dissolved and figs begin to break down, 4 to 5 minutes. Remove saucepan from heat and cool to room temperature, 15 to 20 minutes. Cover saucepan with a lid and refrigerate.
Heat half-and-half in a heavy saucepan over medium-high heat until hot but not boiling, 5 to 6 minutes. Remove saucepan from heat.
Whisk 1/2 cup sugar, egg yolks, and salt together in a bowl until smooth. Temper egg mixture by drizzling 1 to 2 tablespoons half-and-half into egg mixture, while continuously whisking egg mixture until slightly warmed. Pour egg mixture into half-and-half and return saucepan to medium-low heat; cook, whisking constantly, until custard is smooth and thickened, about 5 minutes.
Place saucepan with custard in the refrigerator, stirring occasionally, until chilled, about 30 minutes. Stir sour cream and vanilla extract into chilled custard. Cover saucepan and chill custard completely, at least 3 hours.
Process custard in an ice cream maker according to manufacturer's instructions. Stir fig mixture into ice cream in the last 5 minutes of processing. Transfer fig ice cream to a container and freeze until solid.