Purée figs and dates with water and 2 tablespoons granulated sugar in a blender or food processor until almost smooth.

Whisk flour, anise, baking powder, baking soda, and salt in a bowl. Beat together butter, cream cheese, and remaining 1/2 cup granulated sugar in a large bowl with an electric mixer at moderate speed until pale and fluffy, about 3 minutes. Beat in vanilla and yolk until combined well. Add flour mixture and mix at low speed until just combined.

Halve dough and form each half into a thick rectangle. Chill, wrapped in plastic wrap, until firm, about 1 hour.

Roll out 1 piece of dough between 2 sheets of wax paper into a 9- by 7-inch rectangle, about 1/3 inch thick. Remove top sheet of wax paper and drop half of fig mixture by spoonfuls onto dough, then gently spread in an even layer, leaving a 1/4-inch border around edges. Starting with a long side and using wax paper as an aid, roll up dough jelly-roll style into a log. Roll log in raw sugar to coat completely. Make another log in same manner. Chill logs, wrapped in wax paper, until firm, at least 4 hours or freeze till amost solid.

Preheat oven to 350°F.

Cut logs crosswise into 1/3-inch-thick slices and arrange slices about 2 inches apart on parchment baking sheets. Bake in batches in middle of oven until pale golden, 15 to 17 minutes, then transfer to racks to cool.