Set the strainer over a medium bowl near the stove for the finished ice cream base.
In a 2-3 quart saucepan, whisk the egg yolks, sugar, honey, and salt until blended. Whisk in the milk and cream. Wash the whisk (with soap and water) and set it aside.
Cook the mixture on medium heat (lowering the heat as the mixture approaches temperature), stirring constantly with a heatproof silicon spatula, sweeping the bottom, corners, and sides of the pan constantly to avoid scorching, until the mixture registers 178-180 F on an instant-read thermometer.
Immediately scrape the mixture into the strainer set over the bowl. Stir in the vanilla. Set aside for 10 minutes to cool slightly.
Put the white chocolate into a small bowl. Ladle about 1 cup of the warm ice cream base over the chocolate and leave it for a minute to soften the chocolate. Whisk until the chocolate is completely melted and the mixture is smooth. Scrape the chocolate mixture into the remaining ice cream base and whisk to blend. Set aside to cool. Cover and refrigerate several hours or overnight.
While the base is cooling. Combine the figs, water, honey, sugar, lemon juice, cinnamon, cardamom, and salt in a medium saucepan. Bring the mixture to a simmer. Cover the pan, adjust the heat to maintain a simmer and cook for about 5 minutes. Uncover and continue simmering, stirring frequently at first and constantly at the end, until the liquid is completely reduced and the mixture resembles thick, sticky, chunky jam. Scrape into a bowl. Let cool and then refrigerate until needed.
Freeze the ice cream according to the instruction on your machine. Scrape the finished ice cream into a container, alternating it with the figs. Fold the ice cream with figs together briefly. Cover the fig ice cream container and freeze until firm enough to scoop or until needed.