Make the dough: Whisk the flour, baking powder and salt in a medium bowl. Beat the butter, brown sugar and granulated sugar in a large bowl with a mixer on medium-high speed until fluffy, about 3 minutes. Add the egg and vanilla and beat until combined. Reduce the mixer speed to low; add the flour mixture and beat until just combined. Divide the dough in half and shape into two 2-by-6-inch rectangles. Wrap each rectangle in plastic wrap and refrigerate until firm, at least 1 hour.
Meanwhile, make the filling: Combine the dried figs, jam and juice in a small saucepan. Bring to a simmer over medium-low heat and cook, stirring frequently, until the juice is syrupy, about 25 minutes. Let cool slightly, then transfer to a food processor and puree until smooth. Set aside to cool completely.
Line a baking sheet with parchment paper. Lightly flour 1 piece of dough and roll it out between 2 sheets of parchment paper into a 4-by-12-inch rectangle. Repeat with the remaining dough. Refrigerate until firm, about 30 minutes.
Remove the top pieces of parchment and trim the edges of the dough rectangles. Spread the filling lengthwise down the center of each dough rectangle in a 2-inch-wide strip. Brush the long edges lightly with the beaten egg. Fold 1 long side of dough over the filling, then fold over the other long side; press gently to seal. Place both logs seam-side down on the prepared baking sheet, a few inches apart, and refrigerate until firm, about 30 minutes.
Preheat the oven to 375 degrees F. Bake the cookie logs until golden brown, rotating the pan halfway through, about 20 minutes. Let cool 15 minutes on the pan. Transfer the logs to a cutting board; slice into 1-inch pieces and transfer to racks to cool completely.